In 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.

- History
- The weak and strong topologies
- Weak topology with respect to a pairing
- Weak topology induced by the continuous dual space
- Weak convergence
- Other properties
- Weak-* topology
- Properties
- Examples
- Hilbert spaces
- Distributions
- Weak topology induced by the algebraic dual
- Operator topologies
- See also
- References
- Bibliography

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

Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional 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 analogous weak-* convergence.^{ [1] } The weak topology is also called *topologie faible* and *schwache Topologie*.

Let 𝕂 be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications 𝕂 will be either the field of complex numbers or the field of real numbers with the familiar topologies.

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 (*X*, *Y*, *b*) is a pairing of vector spaces over a topological field 𝕂 (i.e. X and Y are vector spaces over 𝕂 and *b* : *X*×*Y*→ 𝕂 is a bilinear map).

**Notation.**For all*x*∈*X*, let*b*(*x*, •) :*Y*→ 𝕂 denote the linear functional on Y defined by*y*↦*b*(*x*,*y*). Similarly, for all*y*∈*Y*, let*b*(•,*y*) :*X*→ 𝕂 be defined by*x*↦*b*(*x*,*y*).

**Definition.**The**weak topology on X**induced by Y (and b) is the weakest topology on X, denoted by 𝜎(*X*,*Y*,*b*) or simply 𝜎(*X*,*Y*), making all maps*b*(•,*y*) :*X*→ 𝕂 continuous, as y ranges over Y.^{ [1] }

The weak topology on Y is now automatically defined as described in the article Dual system. However, for clarity, we now repeat it.

**Definition.**The**weak topology on Y**induced by X (and b) is the weakest topology on Y, denoted by 𝜎(*Y*,*X*,*b*) or simply 𝜎(*Y*,*X*), making all maps*b*(*x*, •) :*Y*→ 𝕂 continuous, as x ranges over X.^{ [1] }

If the field 𝕂 has an absolute value |⋅|, then the weak topology 𝜎(*X*, *Y*, *b*) on X is induced by the family of seminorms, *p*_{y} : *X*→ ℝ, defined by

*p*_{y}(*x*) := |*b*(*x*,*y*)|

for all *y*∈*Y* and *x*∈*X*. This shows that weak topologies are locally convex.

**Assumption.**We will henceforth assume that 𝕂 is either the real numbers ℝ or the complex numbers ℂ.

We now consider the special case where Y is a vector subspace of the algebraic dual space of X (i.e. a vector space of linear functionals on X).

There is a pairing, denoted by or , called the canonical pairing whose bilinear map is the **canonical evaluation map**, defined by for all and . Note in particular that is just another way of denoting i.e. .

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

In this case, the **weak topology on X** (resp. the **weak topology on Y**), denoted by 𝜎(

The topology σ(*X*,*Y*) is the initial topology of X with respect to Y.

If Y is a vector space of linear functionals on X, then the continuous dual of X with respect to the topology σ(*X*,*Y*) is precisely equal to Y.^{ [1] }( Rudin 1991 , Theorem 3.10)

Let X be a topological vector space (TVS) over 𝕂, that is, X is a 𝕂 vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that X starts with the **original**, **starting**, or **given 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 on X using the topological or continuous dual space , which consists of all linear functionals from X into the base field 𝕂 that are continuous with respect to the given topology.

Recall that is the canonical evaluation map defined by for all and , where in particular, .

**Definition.**The**weak topology on X**is the weak topology on X with respect to the canonical pairing . That is, it is the weakest topology on X making all maps continuous, as ranges over .^{ [1] }

**Definition**: The**weak topology on**is the weak topology on with respect to the canonical pairing . That is, it is the weakest topology on making all maps continuous, as x ranges over X.^{ [1] }This topology is also called the**weak* topology**.

We give alternative definitions below.

Alternatively, the **weak topology** on a TVS X is the initial topology with respect to the family . In other words, it is the coarsest topology on X such that each element of remains a continuous function.

A subbase for the weak topology is the collection of sets of the form where and U is an open subset of the base field 𝕂. In other words, a subset of X 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 .

From this point of view, the weak topology is the coarsest polar topology; see weak topology (polar topology) for details.

The weak topology is characterized by the following condition: a net in X converges in the weak topology to the element x of X if and only if converges to in ℝ or ℂ for all .

In particular, if is a sequence in X, then **converges weakly to**x if

as *n*→∞ for all . In this case, it is customary to write

or, sometimes,

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

If X is a normed space, then the dual space is itself a normed vector space by using the norm

This norm gives rise to a topology, called the **strong topology**, on . This is the topology of uniform 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 a polar topology.

A space X can be embedded into its double dual *X*** by

Thus is an injective linear mapping, though not necessarily surjective (spaces for which *this* canonical embedding is surjective are called reflexive). The **weak-* topology** on is the weak topology induced by the image of . In other words, it is the coarsest topology such that the maps *T _{x}*, defined by from to the base field ℝ or ℂ remain continuous.

- Weak-* convergence

A net in is convergent to in the weak-* topology if it converges pointwise:

for all . In particular, a sequence of converges to provided that

for all *x*∈*X*. In this case, one writes

as *n*→ ∞.

Weak-* convergence is sometimes called the **simple convergence** or the **pointwise convergence**. Indeed, it coincides with the pointwise convergence of linear functionals.

If X is a separable (i.e. has a countable dense subset) locally convex 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.^{ [1] } If X is a separable metrizable locally convex space then the weak* topology on the continuous dual space of X is separable.^{ [1] }

- Properties on normed spaces

By definition, the weak* topology is weaker than the weak topology on . An important fact about the weak* topology is the Banach–Alaoglu theorem: if X is normed, then the closed unit ball in is weak*-compact (more generally, the polar in of a neighborhood of 0 in X is weak*-compact). Moreover, the closed unit ball in a normed space X is compact in the weak topology if and only if X is reflexive.

In more generality, let F be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in , the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak-* topology.

If X is a normed space, then 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, the when X is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of X does not contain any weak* neighborhood of 0.^{ [1] }

If X is a normed space, then X is separable if and only if the weak-* topology on the closed unit ball of is metrizable,^{ [1] } in which case the weak* topology is metrizable on norm-bounded subsets of . If a normed space X has a dual space that is separable (with respect to the dual-norm topology) then X is necessarily separable.^{ [1] } If X is a Banach space, the weak-* topology is not metrizable on all of unless X is finite-dimensional.^{ [2] }

Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space *L*^{2}(ℝ^{n}) . Strong convergence of a sequence to an element ψ means that

as *k*→ ∞. Here the notion of convergence corresponds to the norm on *L*^{2}.

In contrast weak convergence only demands that

for all functions *f*∈*L*^{2} (or, more typically, all *f* in a dense subset of *L*^{2} such as a space of test functions, if the sequence {*ψ*_{k}} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in ℂ.

For example, in the Hilbert space *L*^{2}(0,π), the sequence of functions

form an orthonormal basis. In particular, the (strong) limit of as *k*→ ∞ does not exist. On the other hand, by the Riemann–Lebesgue lemma, the weak limit exists and is zero.

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 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 as *L*^{2}. Thus one is led to consider the idea of a rigged Hilbert space.

Suppose that X is a vector space and *X*^{#} is the algebraic dual space of X (i.e. the vector space of all linear functionals on X). If X is endowed with the weak topology induced by *X*^{#} then the continuous dual space of X is *X*^{#}, every bounded subset of X is contained in a finite-dimensional vector subspace of X, every vector subspace of X is closed and has a topological complement.^{ [3] }

If X and *Y* are topological vector spaces, the space *L*(*X*,*Y*) of continuous linear operators *f* : *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 space *Y* to define operator convergence ( Yosida 1980 , IV.7 Topologies of linear maps). There are, in general, a vast array of possible operator topologies on *L*(*X*,*Y*), whose naming is not entirely intuitive.

For example, the ** strong operator topology ** on *L*(*X*,*Y*) is the topology of *pointwise convergence*. For instance, if *Y* is a normed space, then this topology is defined by the seminorms indexed by *x*∈*X*:

More generally, if a family of seminorms *Q* defines the topology on *Y*, then the seminorms *p*_{q, x} on *L*(*X*,*Y*) defining the strong topology are given by

indexed by *q*∈*Q* and *x*∈*X*.

In particular, see the weak operator topology and weak* operator topology.

In mathematics, more specifically in functional analysis, a **Banach space** 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 vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In quantum mechanics, **bra–ket notation,** or **Dirac notation**, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets". A **ket** looks like "". Mathematically it denotes a vector, , in an abstract (complex) vector space , and physically it represents a state of some quantum system. A **bra** looks like "", and mathematically it denotes a linear form , i.e. a linear map that maps each vector in to a number in the complex plane . Letting the linear functional act on a vector is written as .

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

**Riesz representation theorem**, sometimes called **Riesz–Fréchet representation theorem**, named after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next; a natural isomorphism.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from *X* into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space *X* is reflexive if and only if the canonical evaluation map from *X* into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a *non*-reflexive Banach space that is isometrically isomorphic to its bidual.

In mathematics, a **rigged Hilbert space** is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.

In functional analysis, the **weak operator topology**, often abbreviated **WOT**, is the weakest topology on the set of bounded operators on a Hilbert space , such that the functional sending an operator to the complex number is continuous for any vectors and in the Hilbert space.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis and related areas of mathematics a **polar topology**, **topology of 𝒢-convergence** or **topology of uniform convergence on the sets of**𝒢 is a method to define locally convex topologies on the vector spaces of a pairing.

In functional analysis and related areas of mathematics a **dual topology** is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.

In mathematics, and in particular functional analysis, the **tensor product of Hilbert spaces** is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.

In functional analysis, the concept of a **compact operator on Hilbert space** is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In quantum mechanics, the **expectation value** is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the *most* probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

The mathematical concept of a **Hilbert space**, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.

In mathematics, the **Pettis integral** or **Gelfand–Pettis integral**, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the **weak integral** in contrast to the Bochner integral, which is the strong integral.

In the theory of C*-algebras, the **universal representation** of a C*-algebra is a faithful representation which is the direct sum of the GNS representations corresponding to the states of the C*-algebra. The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. The close relationship between an arbitrary representation of a C*-algebra and its universal representation can be exploited to obtain several criteria for determining whether a linear functional on the algebra is ultraweakly continuous. The method of using the properties of the universal representation as a tool to prove results about the C*-algebra and its representations is commonly referred to as *universal representation techniques* in the literature.

In the field of functional analysis, a subfield of mathematics, a **dual system**, **dual pair**, or a **duality** over a field 𝕂 is a triple (*X*, *Y*, *b*) consisting of two vector spaces over 𝕂 and a bilinear map *b* : *X* × *Y* → 𝕂 such that for all non-zero *x* ∈ *X* the map *y* ↦ *b*(*x*, *y*) is not identically 0 and for all non-zero *y* ∈ *Y*, the map *x* ↦ *b*(*x*, *y*) is not identically 0. The study of dual systems is called **duality theory**.

In functional analysis, the **strong dual** of a topological vector space (TVS) *X* is the continuous dual space of *X* equipped with the **strong topology** or the **topology of uniform convergence on bounded subsets of X**, where this topology is denoted by or . The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, , has the strong dual topology, or may be written.

This is a glossary for the terminology in a mathematical field of functional analysis.

- 1 2 3 4 5 6 7 8 9 10 11 12 13 Narici & Beckenstein 2011, pp. 225-273.
- ↑ Proposition 2.6.12, p. 226 in Megginson, Robert E. (1998),
*An introduction to Banach space theory*, Graduate Texts in Mathematics,**183**, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3 . - ↑ Trèves 2006, pp. 36, 201.

- Conway, John B. (1994),
*A Course in Functional Analysis*(2nd ed.), Springer-Verlag, ISBN 0-387-97245-5 - 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.**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.**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

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