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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 defi ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Fréchet spaces, named after Maurice Fréchet, are special
topological vector spaces 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 als ...
. They are generalizations of
Banach spaces 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 vector ...
( normed vector spaces that are
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
with respect to the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
induced by the norm). All Banach and
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 natural ...
s are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces. A Fréchet space X is defined to be 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 v ...
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 , \infty) s ...
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 als ...
(TVS) that is complete as a TVS, meaning that every
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in X converges to some point in X (see footnote for more details).Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence x_ = \left(x_m\right)_^ in a TVS X is Cauchy if and only if for all neighborhoods U of the origin in X, x_m - x_n \in U whenever m and n are sufficiently large. Note that this definition of a Cauchy sequence does not depend on any particular metric and doesn't even require that X be metrizable. :Important note: Not all authors require that a Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation-invariant
complete metric In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
. Conversely, if the topology of a locally convex space X is induced by a translation-invariant complete metric then X is a Fréchet space. Fréchet was the first to use the term "
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 vector ...
" and Banach in turn then coined the term "Fréchet space" to mean a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
metrizable topological vector space 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 ...
, without the local convexity requirement (such a space is today often called an "
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
"). The condition of locally convex was added later by
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook i ...
. It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "''F''-space" and "Fréchet space" interchangeably. When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "-space" and "Fréchet space" requires local convexity.


Definitions

Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
, the second a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s.


Invariant metric definition

A topological vector space X is a Fréchet space if and only if it satisfies the following three properties:
  1. It is
    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 v ...
    .Some authors do not include local convexity as part of the definition of a Fréchet space.
  2. Its topology be induced by a translation-invariant metric, that is, a metric d : X \times X \to \R such that d(x, y) = d(x + z, y + z) for all x, y, z \in X. This means that a subset U of X is
    open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * Open (Blues Image album), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...
    if and only if for every u \in U there exists an r > 0 such that \ is a subset of U.
  3. Some (or equivalently, every) translation-invariant metric on X inducing the topology of X is
    complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
    . * Assuming that the other two conditions are satisfied, this condition is equivalent to X being a
    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 towards ...
    , meaning that X is a complete
    uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
    when it is endowed with its canonical uniformity (this canonical uniformity is independent of any metric on X and is defined entirely in terms of vector subtraction and X's neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on X is identical to this canonical uniformity).
Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.


Countable family of seminorms definition

The alternative and somewhat more practical definition is the following: a topological vector space X is a Fréchet space if and only if it satisfies the following three properties: # It is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
, # Its topology may be induced by a countable family of seminorms , , \cdot, , _k k = 0, 1, 2, \ldots This means that a subset U \subseteq X is open if and only if for every u \in U there exists K \geq 0 and r > 0 such that \ is a subset of U, # it is complete with respect to the family of seminorms. A family \mathcal P of seminorms on X yields a Hausdorff topology if and only if \bigcap_ \ = \. A sequence x_ = \left(x_n\right)_^ in X converges to x in the Fréchet space defined by a family of seminorms if and only if it converges to x with respect to each of the given seminorms.


As webbed Baire spaces


Comparison to Banach spaces

In contrast to
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 vector ...
s, the complete translation-invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both a total paranorm and an -norm (the stands for Fréchet). Even though the
topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like the open mapping theorem, the
closed graph theorem In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and m ...
, and the Banach–Steinhaus theorem, still hold.


Constructing Fréchet spaces

Recall that a seminorm \, \cdot\, is a function from a vector space X to the real numbers satisfying three properties. For all x, y \in X and all scalars c, \, x\, \geq 0 \, x+y\, \le \, x\, + \, y\, \, c\cdot x\, = , c, \, x\, If \, x\, = 0 \iff x = 0, then \, \cdot\, is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows: To construct a Fréchet space, one typically starts with a vector space X and defines a countable family of seminorms \, \cdot\, _k on X with the following two properties: * if x \in X and \, x\, _k = 0 for all k \geq 0, then x = 0; * if x_ = \left(x_n\right)_^ is a sequence in X which is Cauchy with respect to each seminorm \, \cdot\, _k, then there exists x \in X such that x_ = \left(x_n\right)_^ converges to x with respect to each seminorm \, \cdot\, _k. Then the topology induced by these seminorms (as explained above) turns X into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete. A translation-invariant complete metric inducing the same topology on X can then be defined by d(x,y)=\sum_^\infty 2^\frac \qquad x, y \in X. The function u \mapsto \frac maps [0, \infty) monotonically to [0, 1), and so the above definition ensures that d(x, y) is "small" if and only if there exists K "large" such that \, x - y\, _k is "small" for k = 0, \ldots, K.


Examples


From pure functional analysis

* Every Banach space is a Fréchet space, as the norm induces a translation-invariant metric and the space is complete with respect to this metric. * The Space of real valued sequences, space \R^ of all real valued sequences becomes a Fréchet space if we define the k-th seminorm of a sequence to be the absolute value of the k-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.


From smooth manifolds


From holomorphicity

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the space L^p( , 1 with p < 1. Although this space fails to be locally convex, it is an
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
.


Properties and further notions

If a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them. A Banach space, C^( , b, C^(X, V) with X compact, and H all admit norms, while \R^ and C(\R) do not. A closed subspace of a Fréchet space is a Fréchet space. A quotient of a Fréchet space by a closed subspace is a Fréchet space. The direct sum of a finite number of Fréchet spaces is a Fréchet space. A product of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all for at most countably many of them are trivial (that is, have dimension 0). Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis). So for example, if I \neq \varnothing is any set and X is any non-trivial Fréchet space (such as X = \R for instance), then the product X^I = \prod_ X is a Fréchet space if and only if I is a countable set. Several important tools of functional analysis which are based on the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
remain true in Fréchet spaces; examples are the
closed graph theorem In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and m ...
and the open mapping theorem. The open mapping theorem implies that if \tau \text \tau_2 are topologies on X that make both (X, \tau) and \left(X, \tau_2\right) into
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
metrizable TVSs (such as Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if \tau \subseteq \tau_2 \text \tau_2 \subseteq \tau \text \tau = \tau_2). Every bounded linear operator from a Fréchet space into another
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 als ...
(TVS) is continuous. There exists a Fréchet space X having a bounded subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any bounded subset of M. All Fréchet spaces are
stereotype space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
s. In the theory of stereotype spaces Fréchet spaces are dual objects to Brauner spaces. All
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 , \infty) s ...
Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a Barrelled space, barrelled topo ...
s are separable. A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual converges is strongly convergent. 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 sub ...
X_b^ of a Fréchet space (and more generally, of any metrizable locally convex space) X is a
DF-space In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
.Gabriyelyan, S.S
"On topological spaces and topological groups with certain local countable networks
(2014)
The strong dual of a DF-space is a Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space and a Ptak space. Every Fréchet space is a Ptak space. The strong bidual (that is, 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 sub ...
of the strong dual space) of a metrizable locally convex space is a Fréchet space.


Norms and normability

If X is a locally convex space then the topology of X can be a defined by a family of continuous on X (a norm is a positive-definite
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
) if and only if there exists continuous on X. Even if a Fréchet space has a topology that is defined by a (countable) family of (all norms are also seminorms), then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any single norm). The space of all sequences \mathbb^ (with the product topology) is a Fréchet space. There does not exist any 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 v ...
topology on \mathbb^ that is strictly coarser than this product topology. The space \mathbb^ is not
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 z ...
, which means that its topology can not be defined by any norm. Also, there does not exist
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
norm on \mathbb^. In fact, as the following theorem shows, whenever X is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of \mathbb^ as a subspace. If X is a non-normable Fréchet space on which there exists a continuous norm, then X contains a closed vector subspace that has no topological complement. A metrizable
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 v ...
space 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 z ...
if and only if 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 sub ...
is a Fréchet–Urysohn locally convex space.Gabriyelyan, S.S
"On topological spaces and topological groups with certain local countable networks
(2014)
In particular, if a locally convex metrizable space X (such as a Fréchet space) is normable (which can only happen if X is infinite dimensional) then 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 sub ...
X^_b is not a
Fréchet–Urysohn space In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a spec ...
and consequently, this
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
Hausdorff locally convex space X^_b is also neither metrizable nor normable. 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 sub ...
of a Fréchet space (and more generally, of bornological spaces such as metrizable TVSs) is always 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 ...
and so like any complete TVS, 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 z ...
if and only if its topology can be induced by a complete norm (that is, if and only if it can be made into 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 vector ...
that has the same topology). If X is a Fréchet space then X 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 z ...
if (and only if) there exists a complete norm on its continuous dual space X' such that the norm induced topology on X' is finer than the weak-* topology. Consequently, if a Fréchet space is normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.


Anderson–Kadec theorem

Note that the homeomorphism described in the Anderson–Kadec theorem is necessarily linear.


Differentiation of functions

If X and Y are Fréchet spaces, then the space L(X,Y) consisting of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear maps from X to Y is a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, the Gateaux derivative: Suppose U is an open subset of a Fréchet space X, P : U \to Y is a function valued in a Fréchet space Y, x \in U and h \in X. The map P is differentiable at x in the direction h if the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
D(P)(x)(h) = \lim_ \,\frac\left(P(x + th) - P(x)\right) exists. The map P is said to be continuously differentiable in U if the map D(P) : U \times X \to Y is continuous. Since the product of Fréchet spaces is again a Fréchet space, we can then try to differentiate D(P) and define the higher derivatives of P in this fashion. The derivative operator P : C^( , 1 \to C^( , 1 defined by P(f) = f' is itself infinitely differentiable. The first derivative is given by D(P)(f)(h) = h' for any two elements f, h \in C^( , 1. This is a major advantage of the Fréchet space C^( , 1 over the Banach space C^k( , 1 for finite k. If P : U \to Y is a continuously differentiable function, then the
differential equation 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, ...
x'(t) = P(x(t)),\quad x(0) = x_0\in U need not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces. In general, the inverse function theorem is not true in Fréchet spaces, although a partial substitute is the
Nash–Moser theorem In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required ...
.


Fréchet manifolds and Lie groups

One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like
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 Euclidea ...
\R^n), and one can then extend the concept of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
to these manifolds. This is useful because for a given (ordinary) compact C^ manifold M, the set of all C^
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s f : M \to M forms a generalized Lie group in this sense, and this Lie group captures the symmetries of M. Some of the relations between
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s and Lie groups remain valid in this setting. Another important example of a Fréchet Lie group is the loop group of a compact Lie group G, the smooth (C^) mappings \gamma : S^1 \to G, multiplied pointwise by \left(\gamma_1 \gamma_2\right)(t) = \gamma_1(t) \gamma_2(t)..


Generalizations

If we drop the requirement for the space to be locally convex, we obtain
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
s: vector spaces with complete translation-invariant metrics.
LF-space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces. This means that ''X'' is a direct li ...
s are countable inductive limits of Fréchet spaces.


See also

* * * * * * * * * * * * *


Notes


Citations


References

* * * * * * * * * * * * * * * * * * * {{DEFAULTSORT:Frechet space Topological vector spaces F-spaces