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

Preliminaries

A pairing is a triple (X, Y, b) consisting of two vector spaces over a field \mathbb{K} (either the real numbers or complex numbers) and a bilinear map A dual pair or dual system is a pairing (X, Y, b) satisfying the following two separation axioms:

Polars

The polar or absolute polar of a subset is the set Dually, the polar or absolute polar of a subset is denoted by B^{\circ}, and defined by In this case, the absolute polar of a subset is also called the prepolar of B and may be denoted by The polar is a convex balanced set containing the origin. If then the bipolar of A, denoted by is defined by Similarly, if then the bipolar of B is defined to be

Weak topologies

Suppose that (X, Y, b) is a pairing of vector spaces over \mathbb{K}. The weak topology on X induced by Y (and b) is the weakest TVS topology on X, denoted by or simply making all maps continuous, as y ranges over Y. Similarly, there are the dual definition of the weak topology on Y induced by X (and b), which is denoted by or simply : it is the weakest TVS topology on Y making all maps continuous, as x ranges over X.

Weak boundedness and absorbing polars

It is because of the following theorem that it is almost always assumed that the family \mathcal{G} consists of -bounded subsets of X.

Dual definitions and results

Every pairing (X, Y, b) can be associated with a corresponding pairing where by definition There is a repeating theme in duality theory, which is that any definition for a pairing (X, Y, b) has a corresponding dual definition for the pairing For instance, after defining "X distinguishes points of Y" (resp, "S is a total subset of Y") as above, then the dual definition of "Y distinguishes points of X" (resp, "S is a total subset of X") is immediately obtained. For instance, once is defined then it should be automatically assume that has been defined without mentioning the analogous definition. The same applies to many theorems. In particular, although this article will only define the general notion of polar topologies on Y with \mathcal{G} being a collection of -bounded subsets of X, this article will nevertheless use the dual definition for polar topologies on X with \mathcal{G} being a collection of -bounded subsets of Y. Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous:

Polar topologies

Throughout, (X, Y, b) is a pairing of vector spaces over the field \mathbb{K} and \mathcal{G} is a non-empty collection of -bounded subsets of X. For every and r > 0, is convex and balanced and because G is a -bounded, the set r G^{\circ} is absorbing in Y. The polar topology on Y determined (or generated) by \mathcal{G} (and b), also called the \mathcal{G}-topology on Y or the **topology of uniform convergence on the sets of ** is the unique topological vector space (TVS) topology on Y for which forms a neighbourhood ** sub **basis at the origin. When Y is endowed with this \mathcal{G}-topology then it is denoted by If is a sequence of positive numbers converging to 0 then the defining neighborhood subbasis at 0 may be replaced with without changing the resulting topology. When \mathcal{G} is a directed set with respect to subset inclusion (i.e. if for all there exists some such that ) then the defining neighborhood subbasis at the origin actually forms a neighborhood basis at 0. Every determines a seminorm defined by where and p_G is in fact the Minkowski functional of G^{\circ}. Because of this, the \mathcal{G}-topology on Y is always a locally convex topology. If every positive scalar multiple of a set in \mathcal{G} is contained in some set belonging to \mathcal{G} then the defining neighborhood subbasis at the origin can be replaced with without changing the resulting topology. The following theorem gives ways in which \mathcal{G} can be modified without changing the resulting \mathcal{G}-topology on Y. It is because of this theorem that many authors often require that \mathcal{G} also satisfy the following additional conditions:

<ul> <li>The union of any two sets is contained in some set ;</li> <li>All scalar multiples of every belongs to </li> </ul> Some authors further assume that every x \in X belongs to some set because this assumption suffices to ensure that the \mathcal{G}-topology is Hausdorff. If is a [net](https://bliptext.com/articles/net-mathematics) in Y then in the \mathcal{G}-topology on Y if and only if for every or in words, if and only if for every the [net](https://bliptext.com/articles/net-mathematics) of linear functionals on X converges uniformly to 0 on G; here, for each i \in I, the linear functional is defined by If y \in Y then in the \mathcal{G}-topology on Y if and only if for all A [filter](https://bliptext.com/articles/filters-in-topology) \mathcal{F} on Y converges to an element y \in Y in the \mathcal{G}-topology on Y if \mathcal{F} converges uniformly to y on each

Properties

Throughout, (X, Y, b) is a pairing of vector spaces over the field \mathbb{K} and \mathcal{G} is a non-empty collection of -bounded subsets of X. Proof of (2): If Y ={ 0 } then we're done, so assume otherwise. Since the \mathcal{G}-topology on Y is a TVS topology, it suffices to show that the set { 0 } is closed in Y. Let y \in Y be non-zero, let be defined by for all x \in X, and let Since X distinguishes points of Y, there exists some (non-zero) x \in X such that f(x) \neq 0 where (since f is surjective) it can be assumed without loss of generality that |f(x)| > 1. The set is a -open subset of X that is not empty (since it contains x). Since is a -dense subset of X there exists some and some g \in G such that g \in U. Since g \in U, so that where G^{\circ} is a subbasic closed neighborhood of the origin in the \mathcal{G}-topology on Y. ■

Examples of polar topologies induced by a pairing

Throughout, (X, Y, b) will be a pairing of vector spaces over the field \mathbb{K} and \mathcal{G} will be a non-empty collection of -bounded subsets of X. The following table will omit mention of b. The topologies are listed in an order that roughly corresponds with coarser topologies first and the finer topologies last; note that some of these topologies may be out of order e.g. c(X, Y, b) and the topology below it (i.e. the topology generated by -complete and bounded disks) or if is not Hausdorff. If more than one collection of subsets appears the same row in the left-most column then that means that the same polar topology is generated by these collections.

Weak topology σ(Y, X)

For any x \in X, a basic -neighborhood of x in X is a set of the form: for some real r > 0 and some finite set of points in Y. The continuous dual space of is X, where more precisely, this means that a linear functional f on Y belongs to this continuous dual space if and only if there exists some x \in X such that for all y \in Y. The weak topology is the coarsest TVS topology on Y for which this is true. In general, the convex balanced hull of a -compact subset of Y need not be -compact. If X and Y are vector spaces over the complex numbers (which implies that b is complex valued) then let and Y_{\R} denote these spaces when they are considered as vector spaces over the real numbers \R. Let denote the real part of b and observe that is a pairing. The weak topology on Y is identical to the weak topology This ultimately stems from the fact that for any complex-valued linear functional f on Y with real part then

Mackey topology τ(Y, X)

The continuous dual space of is X (in the exact same way as was described for the weak topology). Moreover, the Mackey topology is the finest locally convex topology on Y for which this is true, which is what makes this topology important. Since in general, the convex balanced hull of a -compact subset of Y need not be -compact, the Mackey topology may be strictly coarser than the topology c(X, Y, b). Since every -compact set is -bounded, the Mackey topology is coarser than the strong topology b(X, Y, b).

Strong topology 𝛽(Y, X)

A neighborhood basis (not just a sub basis) at the origin for the topology is: The strong topology is finer than the Mackey topology.

Polar topologies and topological vector spaces

Throughout this section, X will be a topological vector space (TVS) with continuous dual space X' and will be the canonical pairing, where by definition The vector space X always distinguishes/separates the points of X' but X' may fail to distinguishes the points of X (this necessarily happens if, for instance, X is not Hausdorff), in which case the pairing is not a dual pair. By the Hahn–Banach theorem, if X is a Hausdorff locally convex space then X' separates points of X and thus forms a dual pair.

Properties

<ul> <li>If covers X then the canonical map from X into is well-defined. That is, for all x \in X the evaluation functional on X'. meaning the map is continuous on <li>Suppose that u : E \to F is a continuous linear and that \mathcal{G} and \mathcal{H} are collections of bounded subsets of X and Y, respectively, that each satisfy axioms and Then the [transpose](https://bliptext.com/articles/transpose) of u, is continuous if for every there is some such that <li>If X is a locally convex Hausdorff TVS over the field \mathbb{K} and \mathcal{G} is a collection of bounded subsets of X that satisfies axioms and then the bilinear map defined by is continuous if and only if X is normable and the \mathcal{G}-topology on X' is the strong dual topology b(X', X).</li> <li>Suppose that X is a [Fréchet space](https://bliptext.com/articles/fr-chet-space) and \mathcal{G} is a collection of bounded subsets of X that satisfies axioms and If \mathcal{G} contains all compact subsets of X then is complete.</li> </ul>

Polar topologies on the continuous dual space

Throughout, X will be a TVS over the field \mathbb{K} with continuous dual space X' and X and X' will be associated with the canonical pairing. The table below defines many of the most common polar topologies on X'. The reason why some of the above collections (in the same row) induce the same polar topologies is due to some basic results. A closed subset of a complete TVS is complete and that a complete subset of a Hausdorff and complete TVS is closed. Furthermore, in every TVS, compact subsets are complete and the balanced hull of a compact (resp. totally bounded) subset is again compact (resp. totally bounded). Also, a Banach space can be complete without being weakly complete. If is bounded then B^{\circ} is absorbing in X' (note that being absorbing is a necessary condition for B^{\circ} to be a neighborhood of the origin in any TVS topology on X'). If X is a locally convex space and B^{\circ} is absorbing in X' then B is bounded in X. Moreover, a subset is weakly bounded if and only if S^{\circ} is absorbing in X'. For this reason, it is common to restrict attention to families of bounded subsets of X.

Weak/weak* topology

σ(X{{'}}, X) The topology has the following properties:

<ul> <li>**[Banach–Alaoglu theorem](https://bliptext.com/articles/banach-alaoglu-theorem)**: Every equicontinuous subset of X' is [relatively compact](https://bliptext.com/articles/relatively-compact) for <li>**Theorem** (S. Banach): Suppose that X and Y are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that u : X \to Y is a continuous linear map. Then u is surjective if and only if the transpose of u, is one-to-one and the [image](https://bliptext.com/articles/image-mathematics) of {}^t u is weakly closed in </li> <li>Suppose that X and Y are [Fréchet spaces](https://bliptext.com/articles/fr-chet-space), Z is a Hausdorff locally convex space and that is a separately-continuous bilinear map. Then is continuous. <li> is normable if and only if X is finite-dimensional.</li> <li>When X is infinite-dimensional the topology on X' is strictly coarser than the strong dual topology b(X',X).</li> <li>Suppose that X is a locally convex Hausdorff space and that \hat{X} is its completion. If then is strictly finer than </li> <li>Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the topology.</li> <li>If X is locally convex then a subset is -bounded if and only if there exists a [barrel](https://bliptext.com/articles/barreled-space) B in X such that </li> </ul>

Compact-convex convergence

γ(X{{'}}, X) If X is a Fréchet space then the topologies

Compact convergence

c(X{{'}}, X) If X is a Fréchet space or a LF-space then c(X',X) is complete. Suppose that X is a metrizable topological vector space and that If the intersection of W' with every equicontinuous subset of X' is weakly-open, then W' is open in c(X',X).

Precompact convergence

Banach–Alaoglu theorem: An equicontinuous subset has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on K coincides with the topology.

Mackey topology

τ(X{{big|{{'}}}}, X) By letting \mathcal{G} be the set of all convex balanced weakly compact subsets of X, X' will have the Mackey topology on X' or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by \tau(X', X) and X' with this topology is denoted by

Strong dual topology

b(X{{'}}, X) Due to the importance of this topology, the continuous dual space of X'_b is commonly denoted simply by X''. Consequently, The b(X',X) topology has the following properties:

<ul> <li>If X is locally convex, then this topology is finer than all other \mathcal{G}-topologies on X' when considering only \mathcal{G}'s whose sets are subsets of X.</li> <li>If X is a [bornological space](https://bliptext.com/articles/bornological-space) (e.g. [metrizable](https://bliptext.com/articles/metrizable) or [LF-space](https://bliptext.com/articles/lf-space)) then is complete.</li> <li>If X is a normed space then the strong dual topology on X' may be defined by the norm where x' \in X'.</li> <li>If X is a [LF-space](https://bliptext.com/articles/lf-space) that is the inductive limit of the sequence of space X_k (for ) then is a [Fréchet space](https://bliptext.com/articles/fr-chet-space) if and only if all X_k are normable.</li> <li>If X is a [Montel space](https://bliptext.com/articles/montel-space) then </ul>

Mackey topology

τ(X, X{{big|{{'}}{{'}}}}) By letting be the set of all convex balanced weakly compact subsets of will have the **Mackey topology on X' induced by X or **the topology of uniform convergence on convex balanced weakly compact subsets of X, which is denoted by and X' with this topology is denoted by

<ul> <li>This topology is finer than b(X', X) and hence finer than </li> </ul>

Polar topologies induced by subsets of the continuous dual space

Throughout, X will be a TVS over the field \mathbb{K} with continuous dual space X' and the canonical pairing will be associated with X and X'. The table below defines many of the most common polar topologies on X. The closure of an equicontinuous subset of X' is weak-* compact and equicontinuous and furthermore, the convex balanced hull of an equicontinuous subset is equicontinuous.

Weak topology

𝜎(X, X{{big|{{'}}}}) Suppose that X and Y are Hausdorff locally convex spaces with X metrizable and that u:X\to Y is a linear map. Then u:X\to Y is continuous if and only if is continuous. That is, u : X \to Y is continuous when X and Y carry their given topologies if and only if u is continuous when X and Y carry their weak topologies.

Convergence on equicontinuous sets

𝜀(X, X{{big|{{'}}}}) If was the set of all convex balanced weakly compact equicontinuous subsets of X', then the same topology would have been induced. If X is locally convex and Hausdorff then X's given topology (i.e. the topology that X started with) is exactly That is, for X Hausdorff and locally convex, if E\subset X' then E is equicontinuous if and only if E^{\circ} is equicontinuous and furthermore, for any S is a neighborhood of the origin if and only if S^{\circ} is equicontinuous. Importantly, a set of continuous linear functionals H on a TVS X is equicontinuous if and only if it is contained in the polar of some neighborhood U of the origin in X (i.e. ). Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of X' "encode" all information about X's topology (i.e. distinct TVS topologies on X produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of X".

Mackey topology

τ(X, X{{big|{{'}}}}) Suppose that X is a locally convex Hausdorff space. If X is metrizable or barrelled then X's original topology is identical to the Mackey topology

Topologies compatible with pairings

Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. If \tau is any other locally convex Hausdorff topological vector space topology on X, then \tau is compatible with duality between X and Y if when X is equipped with \tau, then it has Y as its continuous dual space. If X is given the weak topology then is a Hausdorff locally convex topological vector space (TVS) and is compatible with duality between X and Y (i.e. ). The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on X that are compatible with duality between X and Y? The answer to this question is called the Mackey–Arens theorem.