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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 ...
, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of
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 ...
. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
(respectively,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are
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 ...
(respectively, differentiable,
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
, etc.) with respect to the weak topology.


History

Starting in the early 1900s,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
and Marcel Riesz made extensive use of weak convergence. The early pioneers of
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 ...
did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence. The weak topology is also called ''topologie faible'' and ''schwache Topologie''.


The weak and strong topologies

Let \mathbb be a topological field, namely a field with a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
such that addition, multiplication, and division are
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 ...
. In most applications \mathbb will be either the field of complex numbers or the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with the familiar topologies.


Weak topology with respect to a pairing

Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe. The benefit of this more general construction is that any definition or result proved for it applies to ''both'' the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of this more general construction. Suppose is a pairing of vector spaces over a topological field \mathbb (i.e. and are vector spaces over \mathbb and is a bilinear map). :Notation. For all , let denote the linear functional on defined by . Similarly, for all , let be defined by . :Definition. The weak topology on induced by (and ) is the weakest topology on , denoted by or simply , making all maps continuous, as ranges over . The weak topology on is now automatically defined as described in the article Dual system. However, for clarity, we now repeat it. :Definition. The weak topology on induced by (and ) is the weakest topology on , denoted by or simply , making all maps continuous, as ranges over . If the field \mathbb has an
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, then the weak topology on is induced by the 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, , defined by : for all and . This shows that weak topologies are
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 ...
. :Assumption. We will henceforth assume that \mathbb is either the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s \mathbb or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s \mathbb.


Canonical duality

We now consider the special case where is a vector subspace of the algebraic dual space of (i.e. a vector space of linear functionals on ). There is a pairing, denoted by (X,Y,\langle\cdot, \cdot\rangle) or (X,Y), called the canonical pairing whose bilinear map \langle\cdot, \cdot\rangle is the canonical evaluation map, defined by \langle x,x'\rangle =x'(x) for all x\in X and x'\in Y. Note in particular that \langle \cdot,x'\rangle is just another way of denoting x' i.e. \langle \cdot,x'\rangle=x'(\cdot). :Assumption. If is a vector subspace of the algebraic dual space of then we will assume that they are associated with the canonical pairing . In this case, the weak topology on (resp. the weak topology on ), denoted by (resp. by ) is the weak topology on (resp. on ) with respect to the canonical pairing . The topology is the
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
of with respect to . If is a vector space of linear functionals on , then the continuous dual of with respect to the topology is precisely equal to .


The weak and weak* topologies

Let be a topological vector space (TVS) over \mathbb, that is, is a \mathbb
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
so that vector addition and scalar multiplication are continuous. We call the topology that starts with the original, starting, or given topology (the reader is cautioned against using the terms "
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
" and " strong topology" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on using the topological or continuous dual space X^*, which consists of all linear functionals from into the base field \mathbb that are
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 ...
with respect to the given topology. Recall that \langle\cdot,\cdot\rangle is the canonical evaluation map defined by \langle x,x'\rangle =x'(x) for all x\in X and x'\in X^*, where in particular, \langle \cdot,x'\rangle=x'(\cdot)= x'. :Definition. The weak topology on is the weak topology on with respect to the canonical pairing \langle X,X^*\rangle. That is, it is the weakest topology on making all maps x' =\langle\cdot,x'\rangle:X\to\mathbb continuous, as x' ranges over X^*. :Definition: The weak topology on X^* is the weak topology on X^* with respect to the canonical pairing \langle X,X^*\rangle. That is, it is the weakest topology on X^* making all maps \langle x,\cdot\rangle:X^*\to\mathbb continuous, as ranges over . This topology is also called the weak* topology. We give alternative definitions below.


Weak topology induced by the continuous dual space

Alternatively, the weak topology on a TVS is the
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
with respect to the family X^*. In other words, it is the coarsest topology on X such that each element of X^* remains a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
. A subbase for the weak topology is the collection of sets of the form \phi^(U) where \phi\in X^* and is an open subset of the base field \mathbb. In other words, a subset of is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form \phi^(U). From this point of view, the weak topology is the coarsest polar topology.


Weak convergence

The weak topology is characterized by the following condition: a net (x_\lambda) in converges in the weak topology to the element of if and only if \phi(x_\lambda) converges to \phi(x) in \mathbb or \mathbb for all \phi\in X^*. In particular, if x_n is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
in , then x_n converges weakly to if :\varphi(x_n) \to \varphi(x) as for all \varphi \in X^*. In this case, it is customary to write :x_n \overset x or, sometimes, :x_n \rightharpoonup x.


Other properties

If is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and is a
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 topological ...
. If is a normed space, then the dual space X^* is itself a normed vector space by using the norm :\, \phi\, =\sup_ , \phi(x), . This norm gives rise to a topology, called the strong topology, on X^*. This is the topology of
uniform convergence 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 arbitrarily ...
. The uniform and strong topologies are generally different for other spaces of linear maps; see below.


Weak-* topology

The weak* topology is an important example of a polar topology. A space can be embedded into its
double dual 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 co ...
''X**'' by :x \mapsto \begin T_x: X^* \to \mathbb \\ T_x(\phi) = \phi(x) \end Thus T:X\to X^ is an
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 contrapositi ...
linear mapping, though not necessarily
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
(spaces for which ''this'' canonical embedding is surjective are called reflexive). The weak-* topology on X^* is the weak topology induced by the image of T:T(X)\subset X^. In other words, it is the coarsest topology such that the maps ''Tx'', defined by T_x(\phi)=\phi(x) from X^* to the base field \mathbb or \mathbb remain continuous. ;Weak-* convergence A net \phi_ in X^* is convergent to \phi in the weak-* topology if it converges pointwise: :\phi_ (x) \to \phi (x) for all x\in X. In particular, a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of \phi_n\in X^* converges to \phi provided that :\phi_n(x)\to\phi(x) for all . In this case, one writes :\phi_n \overset \phi as . Weak-* convergence is sometimes called the simple convergence or the pointwise convergence. Indeed, it coincides with the
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set an ...
of linear functionals.


Properties

If is a separable (i.e. has a countable dense subset)
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 and ''H'' is a norm-bounded subset of its continuous dual space, then ''H'' endowed with the weak* (subspace) topology is a metrizable topological space. However, for infinite-dimensional spaces, the metric cannot be translation-invariant. If is a separable 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 then the weak* topology on the continuous dual space of is separable. ;Properties on normed spaces By definition, the weak* topology is weaker than the weak topology on X^*. An important fact about the weak* topology is the
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common p ...
: if is normed, then the closed unit ball in X^* is weak*-
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
(more generally, the
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates *Polar climate, the cli ...
in X^* of a neighborhood of 0 in is weak*-compact). Moreover, the closed unit ball in a normed space is compact in the weak topology if and only if is reflexive. In more generality, let be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let be a normed topological vector space over , compatible with the absolute value in . Then in X^*, the topological dual space of continuous -valued linear functionals on , all norm-closed balls are compact in the weak-* topology. If is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded. This implies, in particular, that when is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded). Thus, even though norm-closed balls are compact, X* is not weak*
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
. If is a normed space, then is separable if and only if the weak-* topology on the closed unit ball of X^* is metrizable, in which case the weak* topology is metrizable on norm-bounded subsets of X^*. If a normed space has a dual space that is separable (with respect to the dual-norm topology) then is necessarily separable. If 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 vector ...
, the weak-* topology is not metrizable on all of X^* unless is finite-dimensional.Proposition 2.6.12, p. 226 in .


Examples


Hilbert spaces

Consider, for example, the difference between strong and weak convergence of functions in the
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 ...
. Strong convergence of a sequence \psi_k\in L^2(\R^n) to an element means that :\int_ , \psi_k-\psi , ^2\,\mu\, \to 0 as . Here the notion of convergence corresponds to the norm on . In contrast weak convergence only demands that :\int_ \bar_k f\,\mathrm d\mu \to \int_ \barf\, \mathrm d\mu for all functions (or, more typically, all ''f'' in a dense subset of such as a space of test functions, if the sequence is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in \mathbb. For example, in the Hilbert space , the sequence of functions :\psi_k(x) = \sqrt\sin(k x) form an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
. In particular, the (strong) limit of \psi_k as does not exist. On the other hand, by the Riemann–Lebesgue lemma, the weak limit exists and is zero.


Distributions

One normally obtains spaces of distributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on \mathbb^n). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as . Thus one is led to consider the idea of a rigged Hilbert space.


Weak topology induced by the algebraic dual

Suppose that is a vector space and ''X''# is the algebraic dual space of (i.e. the vector space of all linear functionals on ). If is endowed with the weak topology induced by ''X''# then the continuous dual space of is , every bounded subset of is contained in a finite-dimensional vector subspace of , every vector subspace of is closed and has a topological complement.


Operator topologies

If and are topological vector spaces, the space of continuous linear operators may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space to define operator convergence . There are, in general, a vast array of possible operator topologies on , whose naming is not entirely intuitive. For example, the strong operator topology on is the topology of ''pointwise convergence''. For instance, if is a normed space, then this topology is defined by the seminorms indexed by : :f\mapsto \, f(x)\, _Y. More generally, if a family of seminorms ''Q'' defines the topology on , then the seminorms on defining the strong topology are given by :p_ : f \mapsto q(f(x)), indexed by and . In particular, see the
weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...
and weak* operator topology.


See also

* Eberlein compactum, a compact set in the weak topology *
Weak convergence (Hilbert space) In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology. Definition A sequence of points (x_n) in a Hilbert space ''H'' is said to converge weakly to a point ''x'' in ''H'' if :\langle x ...
*
Weak-star operator topology In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set ''B''(''H'') of bounded operators on a Hilbert space is the weak-* topology obta ...
* Weak convergence of measures *
Topologies on spaces of linear maps In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The ...
*
Topologies on the set of operators on a Hilbert space In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space ...
*
Vague topology In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let X be a locally ...


References


Bibliography

* * * * * * * * * {{Duality and spaces of linear maps General topology Topology Topology of function spaces