Polar topology

The polar or absolute polar of a subset A ⊆ X is the set[1]

${\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{x\in A}|b(x,y)|\leq 1\right\}.}$

Dually, the polar or absolute polar of a subset B ⊆Y is denoted by B° and defined by

${\displaystyle B^{\circ }:=\left\{x\in X:\sup _{y\in B}|b(x,y)|\leq 1\right\}.}$

In this case, the absolute polar of a subset B ⊆Y is also called the prepolar of B and may be denoted by °B.

The polar is a convex balanced set containing the origin.[2]

If A ⊆ X then the bipolar of A, denoted by A°°, is defined by A°° = ${\displaystyle {}^{\circ }}$(A^⊥). Similarly, if B ⊆ Y then the bipolar of B is defined to be B°° = (°B)°.

Suppose that ${\displaystyle (X,Y,b)}$ is a pairing of vector spaces over 𝕂.

Notation: For all x ∈ X, let b(x, •) : Y → 𝕂 denote the linear functional on Y defined by y ↦ b(x, y) and let b(X, •) = { b(x, •) : x ∈ X }.

Similarly, for all y ∈ Y, let b(•, y) : X → 𝕂 be defined by x ↦ b(x, y) and let b(•, Y) = { b(•, y) : y ∈ Y }.

The weak topology on X induced by Y (and b) is the weakest TVS topology on X, denoted by ${\displaystyle \sigma (X,Y,b)}$ or simply ${\displaystyle \sigma (X,Y),}$ making all maps b( •, y) : X → 𝕂 continuous, as y ranges over Y.[3] Similarly, there are the dual definition of the weak topology on Y induced by X (and b), which is denoted by ${\displaystyle \sigma (Y,X,b)}$ or simply ${\displaystyle \sigma (Y,X)}$: it is the weakest TVS topology on Y making all maps b(x, •) : Y → 𝕂 continuous, as x ranges over X.[3]

It is because of the following theorem that it is almost always assumed that the family 𝒢 consists of ${\displaystyle \sigma (X,Y,b)}$-bounded subsets of X.[3]

Every pairing ${\displaystyle (X,Y,b)}$ can be associated with a corresponding pairing ${\displaystyle (Y,X,{\hat {b}})}$ where by definition ${\displaystyle {\hat {b}}(y,x)=b(x,y).}$[3]

There is a repeating theme in duality theory, which is that any definition for a pairing ${\displaystyle (X,Y,b)}$ has a corresponding dual definition for the pairing ${\displaystyle (Y,X,{\hat {b}})}$.

Convention and Definition: Given any definition for a pairing ${\displaystyle (X,Y,b),}$ one obtains a dual definition by applying it to the pairing ${\displaystyle (Y,X,{\hat {b}}).}$ If the definition depends on the order of X and Y (e.g. the definition of "the weak topology ${\displaystyle \sigma (X,Y)}$ defined on X by Y") then by switching the order of X and Y, it is meant that this definition should be applied to ${\displaystyle (Y,X,{\hat {b}})}$ (e.g. this gives us the definition of "the weak topology ${\displaystyle \sigma (Y,X)}$ defined on Y by X").

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 ${\displaystyle \sigma (X,Y)}$ is defined then it should be automatically assume that ${\displaystyle \sigma (Y,X)}$ has been defined without mentioning the analogous definition. The same applies to many theorems.

Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) is given for a pairing ${\displaystyle (X,Y,b)}$ then mention the corresponding dual definition (or result) will be omitted but it may nevertheless be used.

In particular, although this article will only define the general notion of polar topologies on Y with 𝒢 being a collection of ${\displaystyle \sigma (X,Y)}$-bounded subsets of X, this article will nevertheless use the dual definition for polar topologies on X with 𝒢 being a collection of ${\displaystyle \sigma (Y,X)}$-bounded subsets of Y.

Identification of (X, Y) with (Y, X)

Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous:

Convention: This article will use the common practice of treating a pairing ${\displaystyle (X,Y,b)}$ interchangeably with ${\displaystyle \left(Y,X,{\hat {b}}\right)}$ and also denoting ${\displaystyle \left(Y,X,{\hat {b}}\right)}$ by ${\displaystyle (Y,X,b).}$

For any x ∈ X, a basic ${\displaystyle \sigma (Y,X,b)}$-neighborhood of x in X is a set of the form:

${\displaystyle \left\{z\in X:|b(z-x,y_{i})|\leq r{\text{ for all }}i\right\}}$

for some real r > 0 and some finite set of points y₁, ..., y_n in Y.[3]

The continuous dual space of ${\displaystyle (Y,\sigma (Y,X,b))}$ 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 ∈ X such that f (y) = b(x, y) for all y ∈ Y.[3] The weak topology is the coarsest TVS topology on Y for which this is true.

In general, the convex balanced hull of a ${\displaystyle \sigma (Y,X,b)}$-compact subset of Y need not be ${\displaystyle \sigma (Y,X,b)}$-compact.[3]

If X and Y are vector spaces over the complex numbers (which implies that b is complex valued) then let ${\displaystyle X_{\mathbb {R} }}$ and ${\displaystyle Y_{\mathbb {R} }}$ denote these spaces when they are considered as vector spaces over the real numbers ℝ. Let Re b denote the real part of b and observe that (X_ℝ, Y_ℝ, Re b) is a pairing. The weak topology ${\displaystyle \sigma (Y,X,b)}$ on Y is identical to the weak topology 𝜎(X_ℝ, Y_ℝ, Re b). This ultimately stems from the fact that for any complex-valued linear functional f on Y with real part r := Re f, then

f = r (y) - i r (i y)      for all y ∈ Y.

The continuous dual space of ${\displaystyle (Y,\tau (Y,X,b))}$ 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 ${\displaystyle \sigma (Y,X,b)}$-compact subset of Y need not be ${\displaystyle \sigma (Y,X,b)}$-compact,[3] the Mackey topology may be strictly coarser than the topology ${\displaystyle c(X,Y,b).}$ Since every ${\displaystyle \sigma (Y,X,b)}$-compact set is ${\displaystyle \sigma (Y,X,b)}$-bounded, the Mackey topology is coarser than the strong topology ${\displaystyle b(X,Y,b)}$.[3]

A neighborhood basis (not just a subbasis) at the origin for the ${\displaystyle \beta (Y,X,b)}$ topology is:[3]

${\displaystyle \left\{A^{\circ }~:~A\subseteq X{\text{ is a }}\sigma (X,Y,b)-{\text{bounded}}{\text{ subset of }}X\right\}.}$

The strong topology ${\displaystyle \beta (Y,X,b)}$ is finer than the Mackey topology.[3]

• If ∪G ∈ 𝒢 G covers X then the canonical map from X into ${\displaystyle (X'_{\mathcal {G}})'}$ is well-defined. That is, for all x ∈ X the evaluation functional on ${\displaystyle X'.}$ meaning the map ${\displaystyle x'\in X'\mapsto \langle x',x\rangle ,}$ is continuous on ${\displaystyle X'_{\mathcal {G}}.}$
  • If in addition ${\displaystyle X'}$ separates points on X then the canonical map of X into ${\displaystyle (X'_{\mathcal {G}})'}$ is an injection.
• Suppose that u : E → F is a continuous linear and that 𝒢 and ℋ are collections of bounded subsets of X and Y, respectively, that each satisfy axioms 𝒢₁ and 𝒢₂. Then the transpose of u, ${\displaystyle {}^{t}u:Y'_{\mathcal {H}}\to X'_{\mathcal {G}}}$ is continuous if for every G ∈ 𝒢 there is some H ∈ ℋ such that u(G) ⊆ H.[6]
  • In particular, the transpose of u is continuous if ${\displaystyle X'}$ carries the ${\displaystyle \sigma (X',X)}$ (respectively, ${\displaystyle \gamma (X',X),}$ ${\displaystyle c(X',X),}$ ${\displaystyle b(X',X)}$) topology and ${\displaystyle Y'}$ carry any topology stronger than the ${\displaystyle \sigma (Y',Y)}$ topology (respectively, ${\displaystyle \gamma (Y',Y),}$ ${\displaystyle c(Y',Y),}$ ${\displaystyle b(Y',Y)}$).
• If X is a locally convex Hausdorff TVS over the field 𝕂 and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢₁ and 𝒢₂ then the bilinear map ${\displaystyle X\times X'_{\mathcal {G}}\to \mathbb {K} }$ defined by ${\displaystyle (x,x')\mapsto \langle x',x\rangle =x'(x)}$ is continuous if and only if X is normable and the 𝒢-topology on ${\displaystyle X'}$ is the strong dual topology ${\displaystyle b(X',X)}$.
• Suppose that X is a Fréchet space and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢₁ and 𝒢₂. If 𝒢 contains all compact subsets of X then ${\displaystyle X'_{\mathcal {G}}}$ is complete.

The ${\displaystyle \sigma (X',Y)}$ topology has the following properties:

• Banach–Alaoglu theorem: Every equicontinuous subset of ${\displaystyle X'}$ is relatively compact for ${\displaystyle \sigma (X',Y)}$.[9]
  • it follows that the ${\displaystyle \sigma (X',Y)}$-closure of the convex balanced hull of an equicontinuous subset of ${\displaystyle X'}$ is equicontinuous and ${\displaystyle \sigma (X',Y)}$-compact.
• Theorem (S. Banach): Suppose that X and Y are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that ${\displaystyle u:X\to Y}$ is a continuous linear map. Then u is surjective if and only if the transpose of u, ${\displaystyle {}^{t}u:Y'\to X',}$ is one-to-one and the range of ${\displaystyle {}^{t}u}$ is weakly closed in ${\displaystyle X'_{\sigma (X',X)}}$.
• Suppose that X and Y are Fréchet spaces, Z is a Hausdorff locally convex space and that ${\displaystyle u:X'_{\sigma }\times Y'_{\sigma }\to Z'_{\sigma }}$ is a separately-continuous bilinear map. Then ${\displaystyle u:X'_{b}\times Y'_{b}\to Z'_{b}}$ is continuous.
  • In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
• ${\displaystyle X'_{\sigma (X',X)}}$ is normable if and only if X is finite-dimensional.
• When X is infinite-dimensional the ${\displaystyle \sigma (X',Y)}$ topology on ${\displaystyle X'}$ is strictly coarser than the strong dual topology ${\displaystyle b(X',X)}$.
• Suppose that X is a locally convex Hausdorff space and that ${\displaystyle {\hat {X}}}$ is its completion. If ${\displaystyle X\neq {\hat {X}}}$ then ${\displaystyle \sigma (X',{\hat {X}})}$ is strictly finer than ${\displaystyle \sigma (X',Y)}$.
• Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the ${\displaystyle \sigma (X',Y)}$ topology.
• If X is locally convex then a subset H of ${\displaystyle X'}$ is ${\displaystyle \sigma (X',Y)}$-bounded if and only if there exists a barrel B in X such that H ⊆ B°.[3]

If X is a Fréchet space then the topologies γ(X', X) = c(X', X).

If X is a Fréchet space or a LF-space then ${\displaystyle c(X',X)}$ is complete.

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

Banach–Alaoglu theorem: An equicontinuous subset K of ${\displaystyle X'}$ has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on K coincides with the ${\displaystyle \sigma (X',Y)}$ topology.

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

Due to the importance of this topology, the continuous dual space of ${\displaystyle X'_{b}}$ is commonly denoted simply by ${\displaystyle X''.}$ Consequently, ${\displaystyle (X'_{b})'=X''.}$

The ${\displaystyle b(X',X)}$topology has the following properties:

• If X is locally convex, then this topology is finer than all other 𝒢-topologies on ${\displaystyle X'}$ when considering only 𝒢's whose sets are subsets of X.
• If X is a bornological space (ex: metrizable or LF-space) then ${\displaystyle X'_{b(X',X)}}$is complete.
• If X is a normed space then the strong dual topology on ${\displaystyle X'}$ may be defined by the norm ${\displaystyle \|X'\|=\sup _{x\in X,\|x\|=1}|\langle x',x\rangle |,}$ where ${\displaystyle x'\in X'}$.[10]
• If X is a LF-space that is the inductive limit of the sequence of space ${\displaystyle X_{k}}$ (for ${\displaystyle k=0,1\dots }$) then ${\displaystyle X'_{b(X',X)}}$ is a Fréchet space if and only if all ${\displaystyle X_{k}}$ are normable.
• If X is a Montel space then
  • ${\displaystyle X'_{b(X',X)}}$ has the Heine–Borel property (i.e. every closed and bounded subset of ${\displaystyle X'_{b(X',X)}}$ is compact in ${\displaystyle X'_{b(X',X)}}$)
  • On bounded subsets of ${\displaystyle X'_{b(X',X)},}$ the strong and weak topologies coincide (and hence so do all other topologies finer than ${\displaystyle \sigma (X',Y)}$ and coarser than ${\displaystyle b(X',X)}$).
  • Every weakly convergent sequence in ${\displaystyle X'}$ is strongly convergent.

By letting 𝒢'' be the set of all convex balanced weakly compact subsets of ${\displaystyle X''=(X'_{b})',X'}$ will have the Mackey topology on ${\displaystyle X'}$ induced by ${\displaystyle X''}$ or the topology of uniform convergence on convex balanced weakly compact subsets of ${\displaystyle X''}$, which is denoted by ${\displaystyle \tau (X',X'')}$ and ${\displaystyle X'}$ with this topology is denoted by ${\displaystyle X'_{\tau (X',X'')}.}$

• This topology is finer than ${\displaystyle b(X',X)}$ and hence finer than ${\displaystyle \tau (X',X)}$.

Suppose that X and Y are Hausdorff locally convex spaces with X metrizable and that ${\displaystyle u:X\to Y}$ is a linear map. Then ${\displaystyle u:X\to Y}$ is continuous if and only if u : 𝜎(X, X') → 𝜎(Y, Y' is continuous. That is, ${\displaystyle u:X\to Y}$ 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.

If ${\displaystyle {\mathcal {G}}'}$ was the set of all convex balanced weakly compact equicontinuous subsets of ${\displaystyle X',}$ 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 ${\displaystyle \varepsilon (X,X')}$. That is, for X Hausdorff and locally convex, if ${\displaystyle E\subset X'}$ then E is equicontinuous if and only if E° is equicontinuous and furthermore, for any S ⊆ X, S is a neighborhood of 0 if and only if S° 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 0 in X (i.e. H ⊆ U°). 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 ${\displaystyle X'}$ "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".

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 τ(X, X').[11]

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 𝜏 is any other locally convex Hausdorff topological vector space topology on X, then 𝜏 is compatible with duality between X and Y if when X is equipped with 𝜏, then it has Y as its continuous dual space. If X is given the weak topology ${\displaystyle \sigma (X,Y)}$ then X_{𝜎(X, Y)} is a Hausdorff locally convex topological vector space (TVS) and ${\displaystyle \sigma (X,Y)}$ is compatible with duality between X and Y (i.e. ${\displaystyle X_{\sigma (X,Y)}'=\left(X_{\sigma (X,Y)}\right)'=Y}$). 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.