Weakly Closed
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, weak topology is an alternative term for certain initial topologies, often on
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 ...
s or spaces of
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s, for instance on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. The term is most commonly used for the initial topology of a topological vector space (such as a
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
) with respect to its
continuous 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 const ...
. 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, 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 ...
(respectively,
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
,
analytic Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical ...
, 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 and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
and
Marcel Riesz Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford alg ...
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. In 1929,
Banach Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase "Son of man (Judaism), son of man", combining the Hebrew language, Hebrew word ' ...
introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence. The weak topology is called in French and in German.


The weak and strong topologies

Let \mathbb be a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widel ...
, namely a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
with a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 ...
. In most applications \mathbb will be either the field of
complex numbers 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 form a ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 In mathematics, a pairing is an ''R''- bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be '' ...
of vector spaces over a topological field \mathbb (i.e. and are vector spaces over \mathbb and is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for ...
). :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 In mathematics, a dual system, dual pair or a duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces, X and Y, over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb. In mathematics, duality is t ...
. 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 x 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 like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
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 vec ...
. :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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 for ...
s \mathbb.


Canonical duality

We now consider the special case where is a vector subspace of the
algebraic dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
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 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 const ...
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 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 ...
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 strong 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 that ...
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 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) over \mathbb, that is, is a \mathbb
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
equipped with a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
so that vector addition and
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
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 strong 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 that ...
" and "
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the t ...
" 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 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 const ...
X^*, which consists of all
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
s 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 ...
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 strong 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 that ...
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 small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
. A
subbase In topology, a subbase (or subbasis, prebase, prebasis) for the topology of a topological space is a subcollection B of \tau that generates \tau, in the sense that \tau is the smallest topology containing B as open sets. A slightly different de ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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 cal ...
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 vec ...
. 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 as the function domain i ...
. 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, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dua ...
''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 of its codomain; that is, implies (equivalently by contraposition, impl ...
linear mapping, though not necessarily
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
(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 cal ...
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 Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
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 vec ...
space and ''H'' is a norm-bounded subset of its continuous dual space, then ''H'' endowed with the weak* (subspace) topology is a
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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
topological space. However, for infinite-dimensional spaces, the metric cannot be translation-invariant. If is a separable
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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
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 vec ...
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 pro ...
: 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
(more generally, the
polar Polar may refer to: Geography * Geographical pole, either of the two points on Earth where its axis of rotation intersects its surface ** Polar climate, the climate common in polar regions ** Polar regions of Earth, locations within the polar circ ...
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 In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...
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 e ...
. 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 (, ) 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 ...
, 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, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. 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 In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of such as a space of
test function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly suppor ...
s, 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 (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
. 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 In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set ...
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 In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
.


Weak topology induced by the algebraic dual

Suppose that is a vector space and ''X''# is the
algebraic 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 const ...
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 In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space X, is a vector subspace M for which there exists some other vector subspace N of X, called its (topological) complement in X, such that X ...
.


Operator topologies

If and are topological vector spaces, the space of
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...
s 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 In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
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,Ilijas Farah, Combinatorial Set Theory of C*-algebras' (2019), p. 80. is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional ...
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 the 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 :\lim_\l ...
*
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, is a topology on ''B''(''H''), the space of bounded operators on a Hilbert space ''H''. ''B' ...
*
Weak convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures on a space, sharing a com ...
*
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 art ...
*
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 spac ...
* Vague topology


References


Bibliography

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