Inmathematics,weak topology is an alternative term for certaininitial topologies, often ontopological vector spaces or spaces oflinear operators, for instance on aHilbert space. The term is most commonly used for the initial topology of a topological vector space (such as anormed vector space) with respect to itscontinuous dual. The remainder of this article will deal with this case, which is one of the concepts offunctional analysis.

One may call subsets of a topological vector spaceweakly closed (respectively,weakly compact, etc.) if they areclosed (respectively,compact, etc.) with respect to the weak topology. Likewise, functions are sometimes calledweakly continuous (respectively,weakly differentiable,weakly analytic, etc.) if they arecontinuous (respectively,differentiable,analytic, etc.) with respect to the weak topology.

Starting in the early 1900s,David Hilbert andMarcel Riesz made extensive use of weak convergence. The early pioneers offunctional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable.^[1] In 1929,Banach introduced weak convergence for normed spaces and also introduced the analogousweak-* convergence.^[1] The weak topology is calledtopologie faible in French andschwache Topologie in German.

Let${\displaystyle \mathbb{K}}$ be atopological field, namely afield with atopology such that addition, multiplication, and division arecontinuous. In most applications${\displaystyle \mathbb{K}}$ will be either the field ofcomplex numbers or the field ofreal numbers with the familiar topologies.

Both the weak topology and the weak* topology are special cases of a more general construction forpairings, which we now describe. The benefit of this more general construction is that any definition or result proved for it applies toboth 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(X,Y,b) is apairing of vector spaces over a topological field${\displaystyle \mathbb{K}}$ (i.e.X andY are vector spaces over${\displaystyle \mathbb{K}}$ andb :X ×Y →${\displaystyle \mathbb{K}}$ is abilinear map).

Notation. For allx ∈X, letb(x, •) :Y →${\displaystyle \mathbb{K}}$ denote the linear functional onY defined byy ↦b(x,y). Similarly, for ally ∈Y, letb(•,y) :X →${\displaystyle \mathbb{K}}$ be defined byx ↦b(x,y).

Definition. Theweak topology onX induced byY (andb) is the weakest topology onX, denoted by𝜎(X,Y,b) or simply𝜎(X,Y), making all mapsb(•,y) :X →${\displaystyle \mathbb{K}}$ continuous, asy ranges overY.^[1]

The weak topology onY is now automatically defined as described in the articleDual system. However, for clarity, we now repeat it.

Definition. Theweak topology onY induced byX (andb) is the weakest topology onY, denoted by𝜎(Y,X,b) or simply𝜎(Y,X), making all mapsb(x, •) :Y →${\displaystyle \mathbb{K}}$ continuous, asx ranges overX.^[1]

If the field${\displaystyle \mathbb{K}}$ has anabsolute value|⋅|, then the weak topology𝜎(X,Y,b) onX is induced by the family ofseminorms,p_y :X →${\displaystyle \mathbb{R}}$, defined by

p_y(x) := |b(x,y)|

for ally ∈Y andx ∈X. This shows that weak topologies arelocally convex.

Assumption. We will henceforth assume that${\displaystyle \mathbb{K}}$ is either thereal numbers${\displaystyle \mathbb{R}}$ or thecomplex numbers${\displaystyle \mathbb{C}}$.

We now consider the special case whereY is a vector subspace of thealgebraic dual space ofX (i.e. a vector space of linear functionals onX).

There is a pairing, denoted by${\displaystyle (X,Y,⟨\cdot ,\cdot ⟩)}$ or${\displaystyle (X,Y)}$, called thecanonical pairing whose bilinear map${\displaystyle ⟨\cdot ,\cdot ⟩}$ is thecanonical evaluation map, defined by${\displaystyle ⟨x,x^{\prime }⟩=x^{\prime }(x)}$ for all${\displaystyle x\in X}$ and${\displaystyle x^{\prime }\in Y}$. Note in particular that${\displaystyle ⟨\cdot ,x^{\prime }⟩}$ is just another way of denoting${\displaystyle x^{\prime }}$ i.e.${\displaystyle ⟨\cdot ,x^{\prime }⟩=x^{\prime }(\cdot )}$.

Assumption. IfY is a vector subspace of thealgebraic dual space ofX then we will assume that they are associated with the canonical pairing⟨X,Y⟩.

In this case, theweak topology onX (resp. theweak topology onY), denoted by𝜎(X,Y) (resp. by𝜎(Y,X)) is theweak topology onX (resp. onY) with respect to the canonical pairing⟨X,Y⟩.

The topologyσ(X,Y) is theinitial topology ofX with respect toY.

IfY is a vector space of linear functionals onX, then the continuous dual ofX with respect to the topologyσ(X,Y) is precisely equal toY.^[1](Rudin 1991, Theorem 3.10)

LetX be atopological vector space (TVS) over${\displaystyle \mathbb{K}}$, that is,X is a${\displaystyle \mathbb{K}}$vector space equipped with atopology so that vector addition andscalar multiplication are continuous. We call the topology thatX starts with theoriginal,starting, orgiven topology (the reader is cautioned against using the terms "initial topology" 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 onX using the topological orcontinuous dual space${\displaystyle X^{\ast }}$, which consists of alllinear functionals fromX into the base field${\displaystyle \mathbb{K}}$ that arecontinuous with respect to the given topology.

Recall that${\displaystyle ⟨\cdot ,\cdot ⟩}$ is the canonical evaluation map defined by${\displaystyle ⟨x,x^{\prime }⟩=x^{\prime }(x)}$ for all${\displaystyle x\in X}$ and${\displaystyle x^{\prime }\in X^{\ast }}$, where in particular,${\displaystyle ⟨\cdot ,x^{\prime }⟩=x^{\prime }(\cdot )=x^{\prime }}$.

Definition. Theweak topology onX is the weak topology onX with respect to thecanonical pairing${\displaystyle ⟨X,X^{\ast }⟩}$. That is, it is the weakest topology onX making all maps${\displaystyle x^{\prime }=⟨\cdot ,x^{\prime }⟩:X\to \mathbb{K}}$ continuous, as${\displaystyle x^{\prime }}$ ranges over${\displaystyle X^{\ast }}$.^[1]

Definition: Theweak topology on${\displaystyle X^{\ast }}$ is the weak topology on${\displaystyle X^{\ast }}$ with respect to thecanonical pairing${\displaystyle ⟨X,X^{\ast }⟩}$. That is, it is the weakest topology on${\displaystyle X^{\ast }}$ making all maps${\displaystyle ⟨x,\cdot ⟩:X^{\ast }\to \mathbb{K}}$ continuous, asx ranges overX.^[1] This topology is also called theweak* topology.

We give alternative definitions below.

Alternatively, theweak topology on a TVSX is theinitial topology with respect to the family${\displaystyle X^{\ast }}$. In other words, it is thecoarsest topology on X such that each element of${\displaystyle X^{\ast }}$ remains acontinuous function.

Asubbase for the weak topology is the collection of sets of the form${\displaystyle {\varphi }^{-1}(U)}$ where${\displaystyle \varphi \in X^{\ast }}$ andU is an open subset of the base field${\displaystyle \mathbb{K}}$. In other words, a subset ofX 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${\displaystyle {\varphi }^{-1}(U)}$.

From this point of view, the weak topology is the coarsestpolar topology.

The weak topology is characterized by the following condition: anet${\displaystyle (x_{\lambda })}$ inX converges in the weak topology to the elementx ofX if and only if${\displaystyle \varphi (x_{\lambda })}$ converges to${\displaystyle \varphi (x)}$ in${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C}}$ for all${\displaystyle \varphi \in X^{\ast }}$.

In particular, if${\displaystyle x_n}$ is asequence inX, then${\displaystyle x_n}$converges weakly tox if

${\displaystyle \phi (x_n)\to \phi (x)}$

asn → ∞ for all${\displaystyle \phi \in X^{\ast }}$. In this case, it is customary to write

${\displaystyle x_n\stackrel{\mathrm{w}}{⟶}x}$

or, sometimes,

${\displaystyle x_n\rightharpoonup x.}$

IfX is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, andX is alocally convex topological vector space.

IfX is a normed space, then the dual space${\displaystyle X^{\ast }}$ is itself a normed vector space by using the norm

${\displaystyle \Vert \varphi \Vert =\underset{\Vert x\Vert \le 1}{sup}|\varphi (x)|.}$

This norm gives rise to a topology, called thestrong topology, on${\displaystyle X^{\ast }}$. This is the topology ofuniform convergence. The uniform and strong topologies are generally different for other spaces of linear maps; see below.

The weak* topology is an important example of apolar topology.

A spaceX can be embedded into itsdouble dualX** by

${\displaystyle x\mapsto \{\begin{array}{l}{T}_x:X^{\ast }\to \mathbb{K}\\ T_x(\varphi )=\varphi (x)\end{array}}$

Thus${\displaystyle T:X\to X^{\ast \ast }}$ is aninjective linear mapping, though not necessarilysurjective (spaces for whichthis canonical embedding is surjective are calledreflexive). Theweak-* topology on${\displaystyle X^{\ast }}$ is the weak topology induced by the image of${\displaystyle T:T(X)\subset X^{\ast \ast }}$. In other words, it is the coarsest topology such that the mapsT_x, defined by${\displaystyle T_x(\varphi )=\varphi (x)}$ from${\displaystyle X^{\ast }}$ to the base field${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C}}$ remain continuous.

Weak-* convergence

Anet${\displaystyle {\varphi }_{\lambda }}$ in${\displaystyle X^{\ast }}$ is convergent to${\displaystyle \varphi }$ in the weak-* topology if it converges pointwise:

${\displaystyle {\varphi }_{\lambda }(x)\to \varphi (x)}$

for all${\displaystyle x\in X}$. In particular, asequence of${\displaystyle {\varphi }_n\in X^{\ast }}$ converges to${\displaystyle \varphi }$ provided that

${\displaystyle {\varphi }_n(x)\to \varphi (x)}$

for allx ∈X. In this case, one writes

${\displaystyle {\varphi }_n\stackrel{w^{\ast }}{\to }\varphi }$

asn → ∞.

Weak-* convergence is sometimes called thesimple convergence or thepointwise convergence. Indeed, it coincides with thepointwise convergence of linear functionals.

IfX is aseparable (i.e. has a countable dense subset)locally convex space andH is a norm-bounded subset of its continuous dual space, thenH endowed with the weak* (subspace) topology is ametrizable topological space.^[1] However, for infinite-dimensional spaces, the metric cannot be translation-invariant.^[2] IfX is a separablemetrizablelocally convex space then the weak* topology on the continuous dual space ofX is separable.^[1]

Properties on normed spaces

By definition, the weak* topology is weaker than the weak topology on${\displaystyle X^{\ast }}$. An important fact about the weak* topology is theBanach–Alaoglu theorem: ifX is normed, then the closed unit ball in${\displaystyle X^{\ast }}$ is weak*-compact (more generally, thepolar in${\displaystyle X^{\ast }}$ of a neighborhood of 0 inX is weak*-compact). Moreover, the closed unit ball in a normed spaceX is compact in the weak topology if and only ifX isreflexive.

In more generality, letF be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). LetX be a normed topological vector space overF, compatible with the absolute value inF. Then in${\displaystyle X^{\ast }}$, the topological dual spaceX of continuousF-valued linear functionals onX, all norm-closed balls are compact in the weak* topology.

IfX is a normed space, a version of theHeine-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.^[1] This implies, in particular, that whenX is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space ofX does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded).^[1] Thus, even though norm-closed balls are compact, X* is not weak*locally compact.

IfX is a normed space, thenX is separable if and only if the weak* topology on the closed unit ball of${\displaystyle X^{\ast }}$ is metrizable,^[1] in which case the weak* topology is metrizable on norm-bounded subsets of${\displaystyle X^{\ast }}$. If a normed spaceX has a dual space that is separable (with respect to the dual-norm topology) thenX is necessarily separable.^[1] IfX is aBanach space, the weak* topology is not metrizable on all of${\displaystyle X^{\ast }}$ unlessX is finite-dimensional.^[3]

Consider, for example, the difference between strong and weak convergence of functions in theHilbert spaceL²(${\displaystyle {\mathbb{R}}^n}$). Strong convergence of a sequence${\displaystyle {\psi }_k\in L^2({\mathbb{R}}^n)}$ to an elementψ means that

${\displaystyle {\int }_{{\mathbb{R}}^n}|{\psi }_k-\psi {|}^2\mathrm{d}\mu \to 0}$

ask → ∞. Here the notion of convergence corresponds to the norm onL².

In contrast weak convergence only demands that

${\displaystyle {\int }_{{\mathbb{R}}^n}{\overline{\psi }}_{k}f\mathrm{d}\mu \to {\int }_{{\mathbb{R}}^n}\overline{\psi }f\mathrm{d}\mu }$

for all functionsf ∈L² (or, more typically, allf in adense subset ofL² such as a space oftest functions, if the sequence {ψ_k} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in${\displaystyle \mathbb{C}}$.

For example, in the Hilbert spaceL²(0,π), the sequence of functions

${\displaystyle {\psi }_k(x)=\sqrt{2/\pi }\sin (kx)}$

form anorthonormal basis. In particular, the (strong) limit of${\displaystyle {\psi }_k}$ ask → ∞ does not exist. On the other hand, by theRiemann–Lebesgue lemma, the weak limit exists and is zero.

One normally obtains spaces ofdistributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on${\displaystyle {\mathbb{R}}^n}$). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such asL². Thus one is led to consider the idea of arigged Hilbert space.

Suppose thatX is a vector space andX^# is thealgebraic dual space ofX (i.e. the vector space of all linear functionals onX). IfX is endowed with the weak topology induced byX^# then the continuous dual space ofX isX^#, every bounded subset ofX is contained in a finite-dimensional vector subspace ofX, every vector subspace ofX is closed and has atopological complement.^[4]

IfX andY are topological vector spaces, the spaceL(X,Y) ofcontinuous linear operatorsf :X → Y 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 spaceY to define operator convergence (Yosida 1980, IV.7 Topologies of linear maps). There are, in general, a vast array of possibleoperator topologies onL(X,Y), whose naming is not entirely intuitive.

For example, thestrong operator topology onL(X,Y) is the topology ofpointwise convergence. For instance, ifY is a normed space, then this topology is defined by the seminorms indexed byx ∈X:

${\displaystyle f\mapsto \Vert f(x){\Vert }_Y.}$

More generally, if a family of seminormsQ defines the topology onY, then the seminormsp_q,x onL(X,Y) defining the strong topology are given by

${\displaystyle p_{q,x}:f\mapsto q(f(x)),}$

indexed byq ∈Q andx ∈X.

In particular, see theweak operator topology andweak* operator topology.

• Eberlein compactum, a compact set in the weak topology
• Weak convergence (Hilbert space)
• Weak-star operator topology
• Weak convergence of measures
• Topologies on spaces of linear maps
• Topologies on the set of operators on a Hilbert space
• Vague topology

• Conway, John B. (1994),A Course in Functional Analysis (2nd ed.), Springer-Verlag,ISBN 0-387-97245-5
• Folland, G.B. (1999).Real Analysis: Modern Techniques and Their Applications (Second ed.). John Wiley & Sons, Inc.ISBN 978-0-471-31716-6.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Pedersen, Gert (1989),Analysis Now, Springer,ISBN 0-387-96788-5
• Rudin, Walter (1991).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
• Willard, Stephen (February 2004).General Topology. Courier Dover Publications.ISBN 9780486434797.
• Yosida, Kosaku (1980),Functional analysis (6th ed.), Springer,ISBN 978-3-540-58654-8

• General topology
• Topology
• Topology of function spaces

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