Review of Polar Topologies
Review of Polar Topologies
Let $(E, F)$ be a dual pair.
- Recall from the Criteria for a Subset A to be σ(E, F)-Weakly Bounded page that the following statements for a subset $A$ of $E$ to be $\sigma (E, F)$-weakly bounded (i.e., bounded with respect to the $\sigma(E, F)$ topology (the weak topology on $E$ determined by $F$):
Equivalent Statements for a Set to be $\sigma(E, F)$-Weakly Bounded | |
(1) | $A$ is $\sigma(E, F)$-weakly bounded, i.e., bounded with respect to the locally convex topology $\sigma(E, F)$. |
(2) | The function $p' : F \to [0, \infty)$ defined for all $y \in F$ by $p'(y) := \sup \{ |\langle x, y \rangle| : x \in A \}$ is a seminorm on $F$. |
(3) | The polar $A^{\circ}$ (in $F$) is an absorbent subset of $F$. |
- On the Polar Topologies page, for a collection $\mathcal A$ of $\sigma(E, F)$-weakly bounded subsets of $E$, we defined the **Topology of $\mathcal A$-Convergence to be the coarsest locally convex topology on $F$ for which the sets $\{ A^{\circ} : A \in \mathcal A \}$ are neighbourhoods of the origin in $F$. This topology is the coarsest topology determined by the set $Q := \{ p_{A^{\circ}} : A \in \mathcal A \}$ of seminorms where for each $A \in \mathcal A$ we have that:
\begin{align} \quad p_{A^{\circ}}(y) := \sup \{ |\langle x, y \rangle | : x \in A \} \end{align}
- (Note that each is a seminorm because each $A \in \mathcal A$ is $\sigma(E, F)$-weakly bounded). We defined a Polar Topology on $F$ to be any topology of $\mathcal A$-convergence on $F$ where $\mathcal A$ is a collection of $\sigma(E, F)$-weakly bounded sets.
- Also recall that from this point forward, $\mathcal A$ will always be a collection of $\sigma(E, F)$-weakly bounded, $\sigma(E, F)$-weakly closed, and absolutely convex sets that satisfies the following additional properties (B1), (B2), and (B3):
Assumptions on the Collection $\mathcal A$ of $\sigma(E, F)$-Weakly Bounded Sets | |
(B1) | If $A, B \in \mathcal A$ then there exists a $C \in \mathcal A$ such that $A \cup B \subseteq C$. |
(B2) | If $A \in \mathcal A$ and $\lambda \in \mathbf{F}$ then $\lambda A \in \mathcal A$. |
(B3) | $\displaystyle{\bigcup_{A \in \mathcal A} A}$ spans $E$. |
- The assumptions (B1) and (B2) guarantee that the collection $\{ A^{\circ} : A \in \mathcal A \}$ forms a base of neighbourhoods for the topology of $\mathcal A$-convergence on $F$, and the assumptions (B3) guarantees that the toology of $\mathcal A$-convergence is Hausdorff.
- On the Equicontinuity Classification of Hausdorff Locally Compact Topologies page we proved the following important theorem which classifies Hausdorff locally convex topologies as polar topologies:
If $E$ is a Hausdorff and locally convex topological vector space with Hausdorff locally convex topology $\tau$, and if $\mathcal A$ is the collection of all $\tau$-equicontinous subsets of linear forms on $E$, then $\tau$ is the topology of $\mathcal A$-convergence.
- On the Weak Continuity Criterion of t for the Continuity of t' page we proved that if $(E, F)$ and $(G, H)$ are dual pairs and $\mathcal A$ is a collection of $\sigma(E, F)$-weakly bounded sets, then:
If $t : E \to G$ is a linear mapping and $t$ is weakly continuous then $t' : H \to F$ is continuous when $F$ is equipped with the topology of $\mathcal A$-convergence and $H$ is equipped with the topology of $t(\mathcal A)$-convergence.
- In particular, on the For Normed Spaces E and F, t is Continuous IFF t' is Continuous page we proved that if $E$ and $F$ are normed spaces (so that $(E, E')$ and $(F, F')$ are dual pairs and so that $E'$ and $F'$ have their own norm topologies) and if $t : E \to F$ is a linear mapping then:
$t :E \to F$ is continuous if and only if $t' : F' \to E'$ is continuous.