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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the spaces of test functions and distributions are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
s (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset U \subseteq \R^n that have compact support. The space of all test functions, denoted by C^\infty_c(U), is endowed with a certain topology, called the , that makes C^\infty_c(U) into a complete Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
TVS. The
strong dual space In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded su ...
of C^\infty_c(U) is called and is denoted by \mathcal^(U) := \left(C^\infty_c(U)\right)^_b, where the "b" subscript indicates that the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of C^\infty_c(U), denoted by \left(C^\infty_c(U)\right)^, is endowed with the
strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
. There are other possible choices for the space of test functions, which lead to other different
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
s of distributions. If U = \R^n then the use of Schwartz functionsThe Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin. as test functions gives rise to a certain subspace of \mathcal^(U) whose elements are called . These are important because they allow the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions \mathcal^(U) and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are subsets of C_c^\infty(U), such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.Except for the trivial (i.e. identically 0) map, which of course is always analytic. Use of analytic test functions leads to Sato's theory of hyperfunctions.


Notation

The following notation will be used throughout this article: * n is a fixed positive integer and U is a fixed non-empty open subset of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
\R^. * \N = \ denotes the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s. * k will denote a non-negative integer or \infty. * If f is a function then \operatorname(f) will denote its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
and the of f, denoted by \operatorname(f), is defined to be the closure of the set \ in \operatorname(f). * For two functions f, g : U \to \Complex, the following notation defines a canonical pairing: \langle f, g\rangle := \int_U f(x) g(x) \,dx. * A of size n is an element in \N^n (given that n is fixed, if the size of multi-indices is omitted then the size should be assumed to be n). The of a multi-index \alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n is defined as \alpha_1+\cdots+\alpha_n and denoted by , \alpha, . Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index \alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n: \begin x^\alpha &= x_1^ \cdots x_n^ \\ \partial^\alpha &= \frac \end We also introduce a partial order of all multi-indices by \beta \geq \alpha if and only if \beta_i \geq \alpha_i for all 1 \leq i\leq n. When \beta \geq \alpha we define their multi-index binomial coefficient as: \binom := \binom \cdots \binom. * \mathbb will denote a certain non-empty collection of compact subsets of U (described in detail below).


Definitions of test functions and distributions

In this section, we will formally define real-valued distributions on . With minor modifications, one can also define complex-valued distributions, and one can replace \R^n with any (
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...
)
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
. Note that for all j, k \in \ and any compact subsets and of , we have: \begin C^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\ C^k(K) &\subseteq C^k(L) && \text K \subseteq L \\ C^k(K) &\subseteq C^j(K) && \text j \leq k \\ C_c^k(U) &\subseteq C^j_c(U) && \text j \leq k \\ C^k(U) &\subseteq C^j(U) && \text j \leq k \\ \end Distributions on are defined to be the continuous linear functionals on C_c^\infty(U) when this vector space is endowed with a particular topology called the . This topology is unfortunately not easy to define but it is nevertheless still possible to characterize distributions in a way so that no mention of the canonical LF-topology is made. Proposition: If is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...
on C_c^\infty(U) then the is a distribution if and only if the following equivalent conditions are satisfied: # For every compact subset K\subseteq U there exist constants C>0 and N\in \N (dependent on K) such that for all f \in C^\infty(K), , T(f), \leq C \sup \. # For every compact subset K\subseteq U there exist constants C>0 and N\in \N such that for all f \in C_c^\infty(U) with support contained in K,See for example . , T(f), \leq C \sup \. # For any compact subset K\subseteq U and any sequence \_^\infty in C^\infty(K), if \_^\infty converges uniformly to zero on K for all multi-indices \alpha, then T(f_i) \to 0. The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
) is limited if no topologies are placed on C_c^\infty(U) and \mathcal(U). To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
s (TVSs) be defined first. First, a ( non-normable) topology on C^\infty(U) will be defined, then every C^\infty(K) will be endowed with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
induced on it by C^\infty(U), and finally the ( non-metrizable) canonical LF-topology on C_c^\infty(U) will be defined. The space of distributions, being defined as the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of C_c^\infty(U), is then endowed with the (non-metrizable)
strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
induced by C_c^\infty(U) and the canonical LF-topology (this topology is a generalization of the usual operator norm induced topology that is placed on the continuous dual spaces of
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...
s). This finally permits consideration of more advanced notions such as convergence of distributions (both sequences nets), various (sub)spaces of distributions, and operations on distributions, including extending differential equations to distributions.


Choice of compact sets K

Throughout, \mathbb will be any collection of compact subsets of U such that (1) U = \bigcup_ K, and (2) for any compact K \subseteq U there exists some K_2 \in \mathbb such that K \subseteq K_2. The most common choices for \mathbb are: * The set of all compact subsets of U, or * A set \left\ where U = \bigcup_^\infty U_i, and for all , \overline_i \subseteq U_ and U_i is a relatively compact non-empty open subset of U (here, "relatively compact" means that the closure of U_i, in either or \R^n, is compact). We make \mathbb into a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
by defining K_1 \leq K_2 if and only if K_1 \subseteq K_2. Note that although the definitions of the subsequently defined topologies explicitly reference \mathbb, in reality they do not depend on the choice of \mathbb; that is, if \mathbb_1 and \mathbb_2 are any two such collections of compact subsets of U, then the topologies defined on C^k(U) and C_c^k(U) by using \mathbb_1 in place of \mathbb are the same as those defined by using \mathbb_2 in place of \mathbb.


Topology on ''C''''k''(''U'')

We now introduce the seminorms that will define the topology on C^k(U). Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used. All of the functions above are non-negative \R-valuedThe image of the compact set K under a continuous \R-valued map (for example, x \mapsto \left, \partial^p f(x)\ for x \in U) is itself a compact, and thus bounded, subset of \R. If K \neq \varnothing then this implies that each of the functions defined above is \R-valued (that is, none of the supremums above are ever equal to \infty). seminorms on C^k(U). As explained in this article, every set of seminorms on a vector space induces a
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
vector topology. Each of the following sets of seminorms \begin A ~:= \quad &\ \\ B ~:= \quad &\ \\ C ~:= \quad &\ \\ D ~:= \quad &\ \end generate the same
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
vector topology on C^k(U) (so for example, the topology generated by the seminorms in A is equal to the topology generated by those in C). With this topology, C^k(U) becomes a locally convex
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
that is
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
. Every element of A \cup B \cup C \cup D is a continuous seminorm on C^k(U). Under this topology, a net (f_i)_ in C^k(U) converges to f \in C^k(U) if and only if for every multi-index p with , p, < k + 1 and every compact K, the net of partial derivatives \left(\partial^p f_i\right)_
converges uniformly In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...
to \partial^p f on K. For any k \in \, any (von Neumann) bounded subset of C^(U) is a relatively compact subset of C^k(U). In particular, a subset of C^\infty(U) is bounded if and only if it is bounded in C^i(U) for all i \in \N. The space C^k(U) is a Montel space if and only if k = \infty. The topology on C^\infty(U) is the superior limit of the subspace topologies induced on C^\infty(U) by the TVSs C^i(U) as ranges over the non-negative integers. A subset W of C^\infty(U) is open in this topology if and only if there exists i\in \N such that W is open when C^\infty(U) is endowed with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
induced on it by C^i(U).


Metric defining the topology

If the family of compact sets \mathbb = \left\ satisfies U = \bigcup_^\infty U_j and \overline_i \subseteq U_ for all i, then a complete translation-invariant metric on C^\infty(U) can be obtained by taking a suitable countable
Fréchet combination In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
of any one of the above families. For example, using the seminorms (r_)_^\infty results in the metric d(f, g) := \sum^\infty_ \frac \frac = \sum^\infty_ \frac \frac. Often, it is easier to just consider seminorms.


Topology on ''C''''k''(''K'')

As before, fix k \in \. Recall that if K is any compact subset of U then C^k(K) \subseteq C^k(U). For any compact subset K \subseteq U, C^k(K) is a closed subspace of the Fréchet space C^k(U) and is thus also a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
. For all compact K, L \subseteq U satisfying K \subseteq L, denote the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
by \operatorname_K^L : C^k(K) \to C^k(L). Then this map is a linear embedding of TVSs (that is, it is a linear map that is also a topological embedding) whose
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
(or "range") is closed in its
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
; said differently, the topology on C^k(K) is identical to the subspace topology it inherits from C^k(L), and also C^k(K) is a closed subset of C^k(L). The
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
of C^\infty(K) relative to C^\infty(U) is empty. If k is finite then C^k(K) is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
with a topology that can be defined by the norm r_K(f) := \sup_ \left(\sup_ \left, \partial^p f(x_0) \\right). And when k = 2, then \,C^k(K) is even a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. The space C^\infty(K) is a distinguished
Schwartz Schwartz may refer to: *Schwartz (surname), a surname (and list of people with the name) *Schwartz (brand), a spice brand *Schwartz's, a delicatessen in Montreal, Quebec, Canada *Schwartz Publishing, an Australian publishing house *"Danny Schwartz" ...
Montel space so if C^\infty(K) \neq \ then it is
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
and thus a Banach space (although like all other C^k(K), it is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
).


Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''

The definition of C^k(K) depends on so we will let C^k(K;U) denote the topological space C^k(K), which by definition is a
topological subspace In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
of C^k(U). Suppose V is an open subset of \R^n containing U and for any compact subset K \subseteq V, let C^k(K; V) is the vector subspace of C^k(V) consisting of maps with support contained in K. Given f \in C_c^k(U), its is by definition, the function I(f) := F : V \to \Complex defined by: F(x) = \begin f(x) & x \in U, \\ 0 & \text, \end so that F \in C^k(V). Let I : C_c^k(U) \to C^k(V) denote the map that sends a function in C_c^k(U) to its trivial extension on . This map is a linear injection and for every compact subset K \subseteq U (where K is also a compact subset of V since K \subseteq U \subseteq V) we have \begin I\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text \\ I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V) \end If is restricted to C^k(K; U) then the following induced linear map is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(and thus a
TVS-isomorphism In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is a ...
): \begin \,& C^k(K; U) && \to \,&& C^k(K;V) \\ & f && \mapsto\,&& I(f) \\ \end and thus the next two maps (which like the previous map are defined by f \mapsto I(f)) are topological embeddings: C^k(K; U) \to C^k(V), \qquad \text \qquad C^k(K; U) \to C_c^k(V), (the topology on C_c^k(V) is the canonical LF topology, which is defined later). Using the injection I : C_c^k(U) \to C^k(V) the vector space C_c^k(U) is canonically identified with its image in C_c^k(V) \subseteq C^k(V) (however, if U \neq V then I : C_c^\infty(U)\to C_c^\infty(V) is a topological embedding when these spaces are endowed with their canonical LF topologies, although it is continuous). Because C^k(K; U) \subseteq C_c^k(U), through this identification, C^k(K; U) can also be considered as a subset of C^k(V). Importantly, the subspace topology C^k(K; U) inherits from C^k(U) (when it is viewed as a subset of C^k(U)) is identical to the subspace topology that it inherits from C^k(V) (when C^k(K; U) is viewed instead as a subset of C^k(V) via the identification). Thus the topology on C^k(K;U) is independent of the open subset of \R^n that contains . This justifies the practice of written C^k(K) instead of C^k(K; U).


Canonical LF topology

Recall that C_c^k(U) denote all those functions in C^k(U) that have compact support in U, where note that C_c^k(U) is the union of all C^k(K) as ranges over \mathbb. Moreover, for every , C_c^k(U) is a dense subset of C^k(U). The special case when k = \infty gives us the space of test functions. This section defines the canonical LF topology as a direct limit. It is also possible to define this topology in terms of its neighborhoods of the origin, which is described afterwards.


Topology defined by direct limits

For any two sets and , we declare that K \leq L if and only if K \subseteq L, which in particular makes the collection \mathbb of compact subsets of into a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
(we say that such a collection is ). For all compact K, L \subseteq U satisfying K \subseteq L, there are
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
s \operatorname_K^L : C^k(K) \to C^k(L)\quad \text \quad \operatorname_K^U : C^k(K) \to C_c^k(U). Recall from above that the map \operatorname_K^L : C^k(K) \to C^k(L) is a topological embedding. The collection of maps \left\ forms a
direct system In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors fr ...
in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topologica ...
s that is
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
by \mathbb (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the pair (C_c^k(U), \operatorname_^U) where \operatorname_^U := \left(\operatorname_K^U\right)_ are the natural inclusions and where C_c^k(U) is now endowed with the (unique)
strongest "Strongest" is a song recorded by Norwegian singer and songwriter Ina Wroldsen. The song was released on 27 October 2017 and has peaked at number 2 in Norway. "Strongest" is Wroldsen's first solo release on Syco Music Syco Music is a defunc ...
locally convex topology making all of the inclusion maps \operatorname_\bullet^U = (\operatorname_K^U)_ continuous.


Topology defined by neighborhoods of the origin

If is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
subset of C_c^k(U), then is a neighborhood of the origin in the canonical LF topology if and only if it satisfies the following condition: Note that any convex set satisfying this condition is necessarily absorbing in C_c^k(U). Since the topology of any
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually the canonical LF topology by declaring that a convex balanced subset is a neighborhood of the origin if and only if it satisfies condition .


Topology defined via differential operators

A is a sum P := \sum_ c_\alpha \partial^\alpha where c_\alpha \in C^\infty(U) and all but finitely many of c_\alpha are identically . The integer \sup \ is called the of the differential operator P. If P is a linear differential operator of order then it induces a canonical linear map C^k(U) \to C^0(U) defined by \phi \mapsto P\phi, where we shall reuse notation and also denote this map by P. For any 1 \leq k \leq \infty, the canonical LF topology on C_c^k(U) is the weakest locally convex TVS topology making all linear differential operators in U of order \,< k + 1 into continuous maps from C_c^k(U) into C_c^0(U).


Properties of the canonical LF topology


=Canonical LF topology's independence from

= One benefit of defining the canonical LF topology as the direct limit of a
direct system In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors fr ...
is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
collection \mathbb of compact sets. And by considering different collections \mathbb (in particular, those \mathbb mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes C_c^k(U) into a Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
strict LF-space (and also a strict LB-space if k \neq \infty), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).If we take \mathbb to be the set of compact subsets of then we can use the universal property of direct limits to conclude that the inclusion \operatorname_K^U : C^k(K) \to C_c^k(U) is a continuous and even that they are topological embedding for every compact subset K \subseteq U. If however, we take \mathbb to be the set of closures of some countable increasing sequence of relatively compact open subsets of having all of the properties mentioned earlier in this in this article then we immediately deduce that C_c^k(U) is a Hausdorff locally convex strict LF-space (and even a strict LB-space when k \neq \infty). All of these facts can also be proved directly without using direct systems (although with more work).


=Universal property

= From the universal property of direct limits, we know that if u : C_c^k(U) \to Y is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
if and only if for every K \in \mathbb, the restriction of to C^k(K) is continuous (or bounded).


=Dependence of the canonical LF topology on

= Suppose is an open subset of \R^n containing U. Let I : C_c^k(U)\to C_c^k(V) denote the map that sends a function in C_c^k(U) to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if) U \neq V then I\left(C_c^\infty(U)\right) is a dense subset of C_c^\infty(V) and I : C_c^\infty(U)\to C_c^\infty(V) is a topological embedding. Consequently, if U \neq V then the transpose of I : C_c^\infty(U)\to C_c^\infty(V) is neither one-to-one nor onto.


=Bounded subsets

= A subset B \subseteq C_c^k(U) is
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
in C_c^k(U) if and only if there exists some K \in \mathbb such that B \subseteq C^k(K) and B is a bounded subset of C^k(K). Moreover, if K \subseteq U is compact and S \subseteq C^k(K) then S is bounded in C^k(K) if and only if it is bounded in C^k(U). For any 0 \leq k \leq \infty, any bounded subset of C_c^(U) (resp. C^(U)) is a relatively compact subset of C_c^k(U) (resp. C^k(U)), where \infty + 1 = \infty.


=Non-metrizability

= For all compact K \subseteq U, the interior of C^k(K) in C_c^k(U) is empty so that C_c^k(U) is of the first category in itself. It follows from Baire's theorem that C_c^k(U) is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
and thus also
normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
(see this footnoteFor any
TVS TVS may refer to: Mathematics * Topological vector space Television * Television Sydney, TV channel in Sydney, Australia * Television South, ITV franchise holder in the South of England between 1982 and 1992 * TVS Television Network, US dis ...
(
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
or otherwise), the notion of completeness depends entirely on a certain so-called "canonical
uniformity Uniformity may refer to: * Distribution uniformity, a measure of how uniformly water is applied to the area being watered * Religious uniformity, the promotion of one state religion, denomination, or philosophy to the exclusion of all other relig ...
" that is defined using the subtraction operation (see the article
Complete topological vector space In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x toward ...
for more details). In this way, the notion of a
complete TVS In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
does not the existence of any metric. However, if the TVS is metrizable and if d is translation-invariant metric on that defines its topology, then is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if (X, d) is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bo ...
. So if a TVS happens to have a topology that can be defined by such a metric then may be used to deduce the completeness of but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS \R^\N, which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on \R^\N).
for an explanation of how the non-metrizable space C_c^k(U) can be complete even though it does not admit a metric). The fact that C_c^\infty(U) is a
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
Montel space makes up for the non-metrizability of C_c^\infty(U) (see this footnote for a more detailed explanation).One reason for giving C_c^\infty(U) the canonical LF topology is because it is with this topology that C_c^\infty(U) and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
s (TVSs) that are particularly similar to finite-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
s: the
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
s (especially
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s) and the
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
Montel
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. differentiability, such as C_c^\infty(U) and C^\infty(U)) end up being nuclear TVSs while TVSs associated with continuous differentiability (such as C^k(K) with compact and k \neq \infty) often end up being non-nuclear spaces, such as Banach spaces.


=Relationships between spaces

= Using the universal property of direct limits and the fact that the natural inclusions \operatorname_K^L : C^k(K) \to C^k(L) are all topological embedding, one may show that all of the maps \operatorname_K^U : C^k(K) \to C_c^k(U) are also topological embeddings. Said differently, the topology on C^k(K) is identical to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
that it inherits from C_c^k(U), where recall that C^k(K)'s topology was to be the subspace topology induced on it by C^k(U). In particular, both C_c^k(U) and C^k(U) induces the same subspace topology on C^k(K). However, this does imply that the canonical LF topology on C_c^k(U) is equal to the subspace topology induced on C_c^k(U) by C^k(U); these two topologies on C_c^k(U) are in fact equal to each other since the canonical LF topology is metrizable while the subspace topology induced on it by C^k(U) is metrizable (since recall that C^k(U) is metrizable). The canonical LF topology on C_c^k(U) is actually than the subspace topology that it inherits from C^k(U) (thus the natural inclusion C_c^k(U)\to C^k(U) is continuous but a topological embedding). Indeed, the canonical LF topology is so
fine Fine may refer to: Characters * Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny'' * Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano Legal terms * Fine (penalty), money to be paid as punishment for an offe ...
that if C_c^\infty(U)\to X denotes some linear map that is a "natural inclusion" (such as C_c^\infty(U)\to C^k(U), or C_c^\infty(U)\to L^p(U), or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions,
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on C_c^\infty(U), the fine nature of the canonical LF topology means that more linear functionals on C_c^\infty(U) end up being continuous ("more" means as compared to a coarser topology that we could have placed on C_c^\infty(U) such as for instance, the subspace topology induced by some C^k(U), which although it would have made C_c^\infty(U) metrizable, it would have also resulted in fewer linear functionals on C_c^\infty(U) being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making C_c^\infty(U) into a
complete TVS In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
).


=Other properties

= * The differentiation map C_c^\infty(U) \to C_c^\infty(U) is a surjective continuous linear operator. * The bilinear multiplication map C^\infty(\R^m) \times C_c^\infty(\R^n) \to C_c^\infty(\R^) given by (f,g)\mapsto fg is continuous; it is however, hypocontinuous.


Distributions

As discussed earlier, continuous linear functionals on a C_c^\infty(U) are known as distributions on . Thus the set of all distributions on is the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of C_c^\infty(U), which when endowed with the
strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
is denoted by \mathcal^(U). We have the canonical
duality pairing Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** ...
between a distribution on and a test function f \in C_c^\infty(U), which is denoted using
angle brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
by \begin \mathcal^(U) \times C_c^\infty(U) \to \R \\ (T, f) \mapsto \langle T, f \rangle := T(f) \end One interprets this notation as the distribution acting on the test function f to give a scalar, or symmetrically as the test function f acting on the distribution .


Characterizations of distributions

Proposition. If is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...
on C_c^\infty(U) then the following are equivalent: # is a distribution; # : is a continuous function. # is continuous at the origin. # is uniformly continuous. # is a
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vecto ...
. # is sequentially continuous. #* explicitly, for every sequence \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to some f \in C_c^\infty(U), \lim_ T\left(f_i\right) = T(f);Even though the topology of C_c^\infty(U) is not metrizable, a linear functional on C_c^\infty(U) is continuous if and only if it is sequentially continuous. # is sequentially continuous at the origin; in other words, maps null sequences to null sequences. #* explicitly, for every sequence \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to the origin (such a sequence is called a ), \lim_ T\left(f_i\right) = 0. #* a is by definition a sequence that converges to the origin. # maps null sequences to bounded subsets. #* explicitly, for every sequence \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to the origin, the sequence \left(T\left(f_i\right)\right)_^\infty is bounded. # maps Mackey convergent null sequences to bounded subsets; #* explicitly, for every Mackey convergent null sequence \left(f_i\right)_^\infty in C_c^\infty(U), the sequence \left(T\left(f_i\right)\right)_^\infty is bounded. #* a sequence f_ = \left(f_i\right)_^\infty is said to be if there exists a divergent sequence r_ = \left(r_i\right)_^\infty \to \infty of positive real number such that the sequence \left(r_i f_i\right)_^\infty is bounded; every sequence that is Mackey convergent to necessarily converges to the origin (in the usual sense). # The kernel of is a closed subspace of C_c^\infty(U). # The graph of is closed. # There exists a continuous seminorm g on C_c^\infty(U) such that , T, \leq g. # There exists a constant C > 0, a collection of continuous seminorms, \mathcal, that defines the canonical LF topology of C_c^\infty(U), and a finite subset \left\ \subseteq \mathcal such that , T, \leq C(g_1 + \cdots g_m);If \mathcal is also a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
under the usual function comparison then we can take the finite collection to consist of a single element.
# For every compact subset K \subseteq U there exist constants C > 0 and N \in \N such that for all f \in C^\infty(K), , T(f), \leq C \sup \. # For every compact subset K\subseteq U there exist constants C_K>0 and N_K\in \N such that for all f \in C_c^\infty(U) with support contained in K, , T(f), \leq C_K \sup \. # For any compact subset K\subseteq U and any sequence \_^\infty in C^\infty(K), if \_^\infty converges uniformly to zero for all multi-indices p, then T(f_i) \to 0. # Any of the statements immediately above (that is, statements 14, 15, and 16) but with the additional requirement that compact set K belongs to \mathbb.


Topology on the space of distributions

The topology of uniform convergence on bounded subsets is also called .In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space X', where if is a
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...
then this strong dual topology is the same as the usual norm-induced topology on X'.
This topology is chosen because it is with this topology that \mathcal^(U) becomes a
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter what dual topology is placed on \mathcal^(U),Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology. a of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net). No matter which topology is chosen, \mathcal^(U) will be a non-
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
,
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
. The space \mathcal^(U) is separable and has the
strong Pytkeev property Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United St ...
Gabriyelyan, S.S. Kakol J., and·Leiderman, A
"The strong Pitkeev property for topological groups and topological vector spaces"
/ref> but it is neither a k-space nor a
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
, which in particular implies that it is not
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
and also that its topology can be defined using only sequences.


Topological properties


Topological vector space categories

The canonical LF topology makes C_c^k(U) into a complete distinguished strict LF-space (and a strict LB-space if and only if k \neq \infty), which implies that C_c^k(U) is a meager subset of itself. Furthermore, C_c^k(U), as well as its
strong dual space In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded su ...
, is a complete Hausdorff
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
barrelled bornological Mackey space. The strong dual of C_c^k(U) is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
if and only if k \neq \infty so in particular, the strong dual of C_c^\infty(U), which is the space \mathcal^(U) of distributions on , is metrizable (note that the weak-* topology on \mathcal^(U) also is not metrizable and moreover, it further lacks almost all of the nice properties that the
strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
gives \mathcal^(U)). The three spaces C_c^\infty(U), C^\infty(U), and the Schwartz space \mathcal(\R^n), as well as the strong duals of each of these three spaces, are complete
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
Montel
bornological space In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
s, which implies that all six of these locally convex spaces are also
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...
reflexive barrelled Mackey spaces. The spaces C^\infty(U) and \mathcal(\R^n) are both distinguished
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
s. Moreover, both C_c^\infty(U) and \mathcal(\R^n) are Schwartz TVSs.


Convergent sequences


=Convergent sequences and their insufficiency to describe topologies

= The strong dual spaces of C^\infty(U) and \mathcal(\R^n) are
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s but not Fréchet-Urysohn spaces. Moreover, neither the space of test functions C_c^\infty(U) nor its strong dual \mathcal^(U) is a
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
(not even an
Ascoli space Ascoli may refer to: Places in Italy *Ascoli Satriano, a town and ''comune'' in the province of Foggia in the Apulia region *Province of Ascoli Piceno, a province of the Marche region ** Ascoli Piceno, a city which is the seat of the province above ...
),Gabriyelyan, Saa
"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"
(2017)
T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36. which in particular implies that their topologies can be defined entirely in terms of convergent sequences. A sequence \left(f_i\right)_^\infty in C_c^k(U) converges in C_c^k(U) if and only if there exists some K \in \mathbb such that C^k(K) contains this sequence and this sequence converges in C^k(K); equivalently, it converges if and only if the following two conditions hold: # There is a compact set K \subseteq U containing the supports of all f_i. # For each multi-index \alpha, the sequence of partial derivatives \partial^\alpha f_ tends
uniformly Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
to \partial^\alpha f. Neither the space C_c^\infty(U) nor its strong dual \mathcal^(U) is a
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
, and consequently, their topologies can be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is enough to define the canonical LF topology on C_c^\infty(U). The same can be said of the strong dual topology on \mathcal^(U).


=What sequences do characterize

= Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of any Montel space, a sequence converges in the
strong dual topology In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually the convergence of a sequence of distributions; this is fine for sequences but it does extend to the convergence of nets of distributions since a net may converge pointwise but fail to converge in the strong dual topology). Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex
bornological space In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
(such as any of the six TVSs mentioned earlier). Then a linear map F : X \to Y into a locally convex space is continuous if and only if it maps null sequencesA is a sequence that converges to the origin. in to bounded subsets of .Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences. More generally, such a linear map F : X \to Y is continuous if and only if it maps Mackey convergent null sequencesA sequence x_ = \left(x_i\right)_^\infty is said to be if there exists a divergent sequence r_ = \left(r_i\right)_^\infty \to \infty of positive real number such that \left(r_i x_i\right)_^\infty is a bounded set in X. to bounded subsets of Y. So in particular, if a linear map F : X \to Y into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs. For every k \in \, C_c^\infty(U) is
sequentially dense In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
in C_c^k(U). Furthermore, \ is a sequentially dense subset of \mathcal^(U) (with its strong dual topology) and also a sequentially dense subset of the strong dual space of C^\infty(U).


=Sequences of distributions

= A sequence of distributions (T_i)_^\infty converges with respect to the weak-* topology on \mathcal^(U) to a distribution if and only if \langle T_i, f \rangle \to \langle T, f \rangle for every test function f \in \mathcal(U). For example, if f_m:\R\to\R is the function f_m(x) = \begin m & \text x \in ,\frac\\ 0 & \text \end and T_m is the distribution corresponding to f_m, then \langle T_m, f \rangle = m \int_0^ f(x)\, dx \to f(0) = \langle \delta, f \rangle as m \to \infty, so T_m \to \delta in \mathcal^(\R). Thus, for large m, the function f_m can be regarded as an approximation of the Dirac delta distribution.


=Other properties

= * The strong dual space of \mathcal^(U) is TVS isomorphic to C_c^\infty(U) via the canonical TVS-isomorphism C_c^\infty(U) \to (\mathcal^(U))'_ defined by sending f \in C_c^\infty(U) to (that is, to the linear functional on \mathcal^(U) defined by sending d \in \mathcal^(U) to d(f)); * On any bounded subset of \mathcal^(U), the weak and strong subspace topologies coincide; the same is true for C_c^\infty(U); * Every weakly convergent sequence in \mathcal^(U) is strongly convergent (although this does not extend to nets).


Localization of distributions


Preliminaries: Transpose of a linear operator

Operations on distributions and spaces of distributions are often defined by means of the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well known in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
.; . For instance, the well-known Hermitian adjoint of a linear operator between
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
). In general the transpose of a continuous linear map A : X \to Y is the linear map ^A : Y' \to X' \qquad \text \qquad ^A(y') := y' \circ A, or equivalently, it is the unique map satisfying \langle y', A(x)\rangle = \left\langle ^A (y'), x \right\rangle for all x \in X and all y' \in Y' (the prime symbol in y' does not denote a derivative of any kind; it merely indicates that y' is an element of the continuous dual space Y'). Since A is continuous, the transpose ^A : Y' \to X' is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details). In the context of distributions, the characterization of the transpose can be refined slightly. Let A : \mathcal(U) \to \mathcal(U) be a continuous linear map. Then by definition, the transpose of A is the unique linear operator A^t : \mathcal'(U) \to \mathcal'(U) that satisfies: \langle ^A(T), \phi \rangle = \langle T, A(\phi) \rangle \quad \text \phi \in \mathcal(U) \text T \in \mathcal'(U). Since \mathcal(U) is dense in \mathcal'(U) (here, \mathcal(U) actually refers to the set of distributions \left\) it is sufficient that the defining equality hold for all distributions of the form T = D_\psi where \psi \in \mathcal(U). Explicitly, this means that a continuous linear map B : \mathcal'(U) \to \mathcal'(U) is equal to ^A if and only if the condition below holds: \langle B(D_\psi), \phi \rangle = \langle ^A(D_\psi), \phi \rangle \quad \text \phi, \psi \in \mathcal(U) where the right hand side equals \langle ^A(D_\psi), \phi \rangle = \langle D_\psi, A(\phi) \rangle = \langle \psi, A(\phi) \rangle = \int_U \psi \cdot A(\phi) \,dx.


Extensions and restrictions to an open subset

Let V \subseteq U be open subsets of \R^n. Every function f \in \mathcal(V) can be from its domain V to a function on U by setting it equal to 0 on the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
U \setminus V. This extension is a smooth compactly supported function called the and it will be denoted by E_ (f). This assignment f \mapsto E_ (f) defines the operator E_ : \mathcal(V) \to \mathcal(U), which is a continuous injective linear map. It is used to canonically identify \mathcal(V) as a vector subspace of \mathcal(U) (although as a
topological subspace In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
). Its transpose ( explained here) \rho_ := ^E_ : \mathcal'(U) \to \mathcal'(V), is called the and as the name suggests, the image \rho_(T) of a distribution T \in \mathcal'(U) under this map is a distribution on V called the restriction of T to V. The defining condition of the restriction \rho_(T) is: \langle \rho_ T, \phi \rangle = \langle T, E_ \phi \rangle \quad \text \phi \in \mathcal(V). If V \neq U then the (continuous injective linear) trivial extension map E_ : \mathcal(V) \to \mathcal(U) is a topological embedding (in other words, if this linear injection was used to identify \mathcal(V) as a subset of \mathcal(U) then \mathcal(V)'s topology would strictly finer than the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
that \mathcal(U) induces on it; importantly, it would be a
topological subspace In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
since that requires equality of topologies) and its range is also dense in its codomain \mathcal(U). Consequently, if V \neq U then the restriction mapping is neither injective nor surjective. A distribution S \in \mathcal'(V) is said to be if it belongs to the range of the transpose of E_ and it is called if it is extendable to \R^n. Unless U = V, the restriction to V is neither
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
nor surjective.


Spaces of distributions

For all 0 < k < \infty and all 1 < p < \infty, all of the following canonical injections are continuous and have an image/range that is a
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
of their codomain: \begin C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^(U) & \to & L_c^p(U) & \to & L_c^1(U) \\ \downarrow & &\downarrow && \downarrow && && && && && \\ C^\infty(U) & \to & C^k(U) & \to & C^0(U) && && && && && \end where the topologies on the LB-spaces L_c^p(U) are the canonical LF topologies as defined below (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in the codomain. Indeed, C_c^\infty(U) is even
sequentially dense In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
in every C_c^k(U). For every 1 \leq p \leq \infty, the canonical inclusion C_c^\infty(U) \to L^p(U) into the normed space L^p(U) (here L^p(U) has its usual norm topology) is a continuous linear injection and the range of this injection is dense in its codomain if and only if p \neq \infty . Suppose that X is one of the LF-spaces C_c^k(U) (for k \in \) or LB-spaces L^p_c(U) (for 1 \leq p \leq \infty) or normed spaces L^p(U) (for 1 \leq p < \infty). Because the canonical injection \operatorname_X : C_c^\infty(U) \to X is a continuous injection whose image is dense in the codomain, this map's
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^\operatorname_X : X'_b \to \mathcal'(U) = \left(C_c^\infty(U)\right)'_b is a continuous injection. This injective transpose map thus allows the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
X' of X to be identified with a certain vector subspace of the space \mathcal'(U) of all distributions (specifically, it is identified with the image of this transpose map). This continuous transpose map is not necessarily a TVS-embedding so the topology that this map transfers from its domain to the image \operatorname\left(^\operatorname_X\right) is finer than the subspace topology that this space inherits from \mathcal^(U). A linear subspace of \mathcal^(U) carrying a locally convex topology that is finer than the subspace topology induced by \mathcal^(U) = \left(C_c^\infty(U)\right)^_b is called . Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order \leq some integer, distributions induced by a positive Radon measure, distributions induced by an L^p-function, etc.) and any representation theorem about the dual space of may, through the transpose ^\operatorname_X : X'_b \to \mathcal^(U), be transferred directly to elements of the space \operatorname \left(^\operatorname_X\right).


Compactly supported ''Lp''-spaces

Given 1 \leq p \leq \infty, the vector space L_c^p(U) of on U and its topology are defined as direct limits of the spaces L_c^p(K) in a manner analogous to how the canonical LF-topologies on C_c^k(U) were defined. For any compact K \subseteq U, let L^p(K) denote the set of all element in L^p(U) (which recall are equivalence class of Lebesgue measurable L^p functions on U) having a representative f whose support (which recall is the closure of \ in U) is a subset of K (such an f is almost everywhere defined in K). The set L^p(K) is a closed vector subspace L^p(U) and is thus a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
and when p = 2, even a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. Let L_c^p(U) be the union of all L^p(K) as K \subseteq U ranges over all compact subsets of U. The set L_c^p(U) is a vector subspace of L^p(U) whose elements are the (equivalence classes of) compactly supported L^p functions defined on U (or almost everywhere on U). Endow L_c^p(U) with the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that ...
(direct limit topology) induced by the inclusion maps L^p(K) \to L_c^p(U) as K \subseteq U ranges over all compact subsets of U. This topology is called the and it is equal to the final topology induced by any countable set of inclusion maps L^p(K_n) \to L_c^p(U) (n = 1, 2, \ldots) where K_1 \subseteq K_2 \subseteq \cdots are any compact sets with union equal to U. This topology makes L_c^p(U) into an LB-space (and thus also an LF-space) with a topology that is strictly finer than the norm (subspace) topology that L^p(U) induces on it.


Radon measures

The inclusion map \operatorname : C_c^\infty(U) \to C_c^0(U) is a continuous injection whose image is dense in its codomain, so the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^\operatorname : \left(C_c^0(U)\right)^_b \to \mathcal^(U) = \left(C_c^\infty(U)\right)^_b is also a continuous injection. Note that the continuous dual space \left(C_c^0(U)\right)^_b can be identified as the space of
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, where there is a one-to-one correspondence between the continuous linear functionals T \in \left(C_c^0(U)\right)^_b and integral with respect to a Radon measure; that is, * if T \in \left(C_c^0(U)\right)^_b then there exists a Radon measure \mu on such that for all f \in C_c^0(U), T(f) = \textstyle \int_U f \, d\mu, and * if \mu is a Radon measure on then the linear functional on C_c^0(U) defined by C_c^0(U) \ni f \mapsto \textstyle \int_U f \, d\mu is continuous. Through the injection ^\operatorname : \left(C_c^0(U)\right)^_b \to \mathcal^(U), every Radon measure becomes a distribution on . If f is a locally integrable function on then the distribution \phi \mapsto \textstyle \int_U f(x) \phi(x) \, dx is a Radon measure; so Radon measures form a large and important space of distributions. The following is the theorem of the structure of distributions of
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s, which shows that every Radon measure can be written as a sum of derivatives of locally L^\infty functions in : Positive Radon measures A linear function on a space of functions is called if whenever a function f that belongs to the domain of is non-negative (meaning that f is real-valued and f \geq 0) then T(f) \geq 0. One may show that every positive linear functional on C_c^0(U) is necessarily continuous (that is, necessarily a Radon measure).
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
is an example of a positive Radon measure.


Locally integrable functions as distributions

One particularly important class of Radon measures are those that are induced locally integrable functions. The function f : U \to \R is called if it is Lebesgue integrable over every compact subset of .For more information on such class of functions, see the entry on locally integrable functions. This is a large class of functions which includes all continuous functions and all
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
L^p functions. The topology on \mathcal(U) is defined in such a fashion that any locally integrable function f yields a continuous linear functional on \mathcal(U) – that is, an element of \mathcal^(U) – denoted here by T_f, whose value on the test function \phi is given by the Lebesgue integral: \langle T_f, \phi \rangle = \int_U f \phi\,dx. Conventionally, one abuses notation by identifying T_f with f, provided no confusion can arise, and thus the pairing between T_f and \phi is often written \langle f, \phi \rangle = \langle T_f, \phi \rangle. If f and are two locally integrable functions, then the associated distributions T_f and are equal to the same element of \mathcal^(U) if and only if f and are equal
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
(see, for instance, ). In a similar manner, every
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
\mu on defines an element of \mathcal^(U) whose value on the test function \phi is \textstyle\int\phi \,d\mu. As above, it is conventional to abuse notation and write the pairing between a Radon measure \mu and a test function \phi as \langle \mu, \phi \rangle. Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure. Test functions as distributions The test functions are themselves locally integrable, and so define distributions. The space of test functions C_c^\infty(U) is sequentially dense in \mathcal^(U) with respect to the strong topology on \mathcal^(U). This means that for any T \in \mathcal^(U), there is a sequence of test functions, (\phi_i)_^\infty, that converges to T \in \mathcal^(U) (in its strong dual topology) when considered as a sequence of distributions. Or equivalently, \langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text \psi \in \mathcal(U). Furthermore, C_c^\infty(U) is also sequentially dense in the strong dual space of C^\infty(U).


Distributions with compact support

The inclusion map \operatorname : C_c^\infty(U) \to C^\infty(U) is a continuous injection whose image is dense in its codomain, so the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^\operatorname : \left(C^\infty(U)\right)^_b \to \mathcal^(U) = \left(C_c^\infty(U)\right)^_b is also a continuous injection. Thus the image of the transpose, denoted by \mathcal^(U), forms a space of distributions when it is endowed with the strong dual topology of \left(C^\infty(U)\right)^_b (transferred to it via the transpose map ^\operatorname : \left(C^\infty(U)\right)^_b \to \mathcal^(U), so the topology of \mathcal^(U) is finer than the subspace topology that this set inherits from \mathcal^(U)). The elements of \mathcal^(U) = \left(C^\infty(U)\right)^_b can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent, * T \in \mathcal^(U); * the support of is compact; * the restriction of T to C_c^\infty(U), when that space is equipped with the subspace topology inherited from C^\infty(U) (a coarser topology than the canonical LF topology), is continuous; * there is a compact subset of such that for every test function \phi whose support is completely outside of , we have T(\phi)=0. Compactly supported distributions define continuous linear functionals on the space C^\infty(U); recall that the topology on C^\infty(U) is defined such that a sequence of test functions \phi_k converges to 0 if and only if all derivatives of \phi_k converge uniformly to 0 on every compact subset of . Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from C_c^\infty(U) to C^\infty(U).


Distributions of finite order

Let k \in \N. The inclusion map \operatorname : C_c^\infty(U) \to C_c^k(U) is a continuous injection whose image is dense in its codomain, so the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^\operatorname : \left(C_c^k(U)\right)^_b \to \mathcal^(U) = \left(C_c^\infty(U)\right)^_b is also a continuous injection. Consequently, the image of ^\operatorname, denoted by \mathcal'^k(U), forms a space of distributions when it is endowed with the strong dual topology of \left(C_c^k(U)\right)^_b (transferred to it via the transpose map ^\operatorname : \left(C^\infty(U)\right)^_b \to \mathcal'^k(U), so \mathcal'^(U)'s topology is finer than the subspace topology that this set inherits from \mathcal^(U)). The elements of \mathcal'^k(U) are The distributions of order \,\leq 0, which are also called are exactly the distributions that are Radon measures (described above). For 0 \neq k \in \N, a is a distribution of order \,\leq k that is not a distribution of order \,\leq k - 1 A distribution is said to be of if there is some integer such that it is a distribution of order \,\leq k, and the set of distributions of finite order is denoted by \mathcal'^(U). Note that if k \leq 1 then \mathcal'^k(U) \subseteq \mathcal'^(U) so that \mathcal'^(U) is a vector subspace of \mathcal^(U) and furthermore, if and only if \mathcal'^(U) = \mathcal^(U). Structure of distributions of finite order Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if \rho_ is the restriction mapping from to , then the image of \mathcal^(U) under \rho_ is contained in \mathcal'^(V). The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all B ...
s: Example. (Distributions of infinite order) Let U := (0, \infty) and for every test function f, let S f := \sum_^\infty (\partial^ f)\left(\frac\right). Then is a distribution of infinite order on . Moreover, can not be extended to a distribution on \R; that is, there exists no distribution on \R such that the restriction of to is equal to .


Tempered distributions and Fourier transform

Defined below are the , which form a subspace of \mathcal^(\R^n), the space of distributions on \R^n. This is a proper subspace: while every tempered distribution is a distribution and an element of \mathcal^(\R^n), the converse is not true. Tempered distributions are useful if one studies the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in \mathcal^(\R^n). Schwartz space The Schwartz space, \mathcal(\R^n), is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus \phi:\R^n\to\R is in the Schwartz space provided that any derivative of \phi, multiplied with any power of , x, , converges to 0 as , x, \to \infty. These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices \alpha and \beta define: p_ (\phi) ~=~ \sup_ \left, x^\alpha \partial^\beta \phi(x) \. Then \phi is in the Schwartz space if all the values satisfy: p_ (\phi) < \infty. The family of seminorms p_ defines a
locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topology on the Schwartz space. For n = 1, the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology: , f, _ = \sup_ \left(\sup_ \left\\right), \qquad k,m \in \N. Otherwise, one can define a norm on \mathcal(\R^n) via \, \phi \, _k ~=~ \max_ \sup_ \left, x^\alpha \partial^\beta \phi(x)\, \qquad k \geq 1. The Schwartz space is a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
(i.e. a complete
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \inf ...
locally convex space). Because the Fourier transform changes \partial^\alpha into multiplication by x^\alpha and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function. A sequence \ in \mathcal(\R^n) converges to 0 in \mathcal(\R^n) if and only if the functions (1 + , x, )^k (\partial^p f_i)(x) converge to 0 uniformly in the whole of \R^n, which implies that such a sequence must converge to zero in C^\infty(\R^n). \mathcal(\R^n) is dense in \mathcal(\R^n). The subset of all analytic Schwartz functions is dense in \mathcal(\R^n) as well. The Schwartz space is
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space * Nuclear ...
and the tensor product of two maps induces a canonical surjective TVS-isomorphisms \mathcal(\R^m) \ \widehat\ \mathcal(\R^n) \to \mathcal(\R^), where \widehat represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product). Tempered distributions The inclusion map \operatorname : \mathcal(\R^n) \to \mathcal(\R^n) is a continuous injection whose image is dense in its codomain, so the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^\operatorname : (\mathcal(\R^n))'_b \to \mathcal^(\R^n) is also a continuous injection. Thus, the image of the transpose map, denoted by \mathcal^(\R^n), forms a space of distributions when it is endowed with the strong dual topology of (\mathcal(\R^n))'_b (transferred to it via the transpose map ^\operatorname : (\mathcal(\R^n))'_b \to \mathcal^(\R^n), so the topology of \mathcal^(\R^n) is finer than the subspace topology that this set inherits from \mathcal^(\R^n)). The space \mathcal^(\R^n) is called the space of . It is the continuous
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if \left(\text \alpha, \beta \in \N^n: \lim_ p_ (\phi_m) = 0\right) \Longrightarrow \lim_ T(\phi_m)=0. The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
L^p(\R^n) for p \geq 1 are tempered distributions. The can also be characterized as , meaning that each derivative of grows at most as fast as some
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of \phi decays faster than every inverse power of , x, . An example of a rapidly falling function is , x, ^n\exp (-\lambda , x, ^\beta) for any positive n, \lambda, \beta. Fourier transform To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform F : \mathcal(\R^n) \to \mathcal(\R^n) is a TVS- automorphism of the Schwartz space, and the is defined to be its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
^F : \mathcal^(\R^n) \to \mathcal^(\R^n), which (abusing notation) will again be denoted by . So the Fourier transform of the tempered distribution is defined by (FT)(\psi) = T(F \psi) for every Schwartz function \psi. FT is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that F \dfrac = ixFT and also with convolution: if is a tempered distribution and \psi is a smooth function on \R^n, \psi T is again a tempered distribution and F(\psi T) = F \psi * FT is the convolution of FT and F \psi. In particular, the Fourier transform of the constant function equal to 1 is the \delta distribution. Expressing tempered distributions as sums of derivatives If T \in \mathcal^(\R^n) is a tempered distribution, then there exists a constant C > 0, and positive integers and such that for all Schwartz functions \phi \in \mathcal(\R^n) \langle T, \phi \rangle \leq C\sum\nolimits_\sup_ \left, x^\alpha \partial^\beta \phi(x) \=C\sum\nolimits_ p_(\phi). This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function and a multi-index \alpha such that T = \partial^\alpha F. Restriction of distributions to compact sets If T \in \mathcal^(\R^n), then for any compact set K \subseteq \R^n, there exists a continuous function compactly supported in \R^n (possibly on a larger set than itself) and a multi-index \alpha such that T = \partial^\alpha F on C_c^\infty(K).


Tensor product of distributions

Let U \subseteq \R^m and V \subseteq \R^n be open sets. Assume all vector spaces to be over the field \mathbb, where \mathbb=\R or \Complex. For f \in \mathcal(U \times V) define for every u \in U and every v \in V the following functions: \begin f_u : \,& V && \to \,&& \mathbb && \quad \text \quad && f^v : \,&& U && \to \,&& \mathbb \\ & y && \mapsto\,&& f(u, y) && && && x && \mapsto\,&& f(x, v) \\ \end Given S \in \mathcal^(U) and T \in \mathcal^(V), define the following functions: \begin \langle S, f^\rangle : \,& V && \to \,&& \mathbb && \quad \text \quad && \langle T, f_\rangle : \,&& U && \to \,&& \mathbb \\ & v && \mapsto\,&& \langle S, f^v \rangle && && && u && \mapsto\,&& \langle T, f_u \rangle \\ \end where \langle T, f_\rangle \in \mathcal(U) and \langle S, f^\rangle \in \mathcal(V). These definitions associate every S \in \mathcal'(U) and T \in \mathcal'(V) with the (respective) continuous linear map: \begin \,& \mathcal(U \times V) && \to \,&& \mathcal(V) && \quad \text \quad && \,&& \mathcal(U \times V) && \to \,&& \mathcal(U) \\ & f && \mapsto\,&& \langle S, f^ \rangle && && && f && \mapsto\,&& \langle T, f_ \rangle \\ \end Moreover, if either S (resp. T) has compact support then it also induces a continuous linear map of C^\infty(U \times V) \to C^\infty(V) (resp. denoted by S \otimes T or T \otimes S, is the distribution in U \times V defined by: (S \otimes T)(f) := \langle S, \langle T, f_ \rangle \rangle = \langle T, \langle S, f^\rangle \rangle.


Schwartz kernel theorem

The tensor product defines a bilinear map \begin \,& \mathcal^(U) \times \mathcal^(V) && \to \,&& \mathcal^(U \times V) \\ & ~~~~~~~~(S, T) && \mapsto\,&& S \otimes T \\ \end the span of the range of this map is a dense subspace of its codomain. Furthermore, \operatorname (S \otimes T) = \operatorname(S) \times \operatorname(T). Moreover (S,T) \mapsto S \otimes T induces continuous bilinear maps: \begin &\mathcal^(U) &&\times \mathcal^(V) &&\to \mathcal^(U \times V) \\ &\mathcal^(\R^m) &&\times \mathcal^(\R^n) &&\to \mathcal^(\R^) \\ \end where \mathcal' denotes the space of distributions with compact support and \mathcal is the Schwartz space of rapidly decreasing functions. This result does not hold for
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s such as L^2 and its dual space. Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space L^2? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because \mathcal(U) is a nuclear space that the Schwartz kernel theorem holds. Like Hilbert spaces, nuclear spaces may be thought as of generalizations of finite dimensional Euclidean space.


Using holomorphic functions as test functions

The success of the theory led to investigation of the idea of hyperfunction, in which spaces of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...
. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.


See also

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Notes


References


Bibliography

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Further reading

* M. J. Lighthill (1959). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge University Press. (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals) * V.S. Vladimirov (2002). ''Methods of the theory of generalized functions''. Taylor & Francis. * . * . * . * . * . {{Topological vector spaces Functional analysis Generalized functions Generalizations of the derivative Smooth functions Topological vector spaces