Review of Polar and Bipolar Sets
Review of Polar and Bipolar Sets
Let $(E, F)$ be a dual pair and let $A \subseteq E$.
- On The Polar of a Set page we defined The Polar of $A$ in $F$ to be the subset $A^{\circ}$ which consists of all points $y \in F$ such that:
\begin{align} \quad \sup \{ |\langle x, y \rangle : x \in A \} \leq 1 \end{align}
- We then examined some useful properties of polars of sets which are summarized below:
Properties of Polar Sets | |
(1) | If $A^{\circ}$ is absolutely convex and $\sigma(F, E)$-closed. |
(2) | If $A \subseteq B$ then $B^{\circ} \subseteq A^{\circ}$. |
(3) | For all $\lambda \in \mathbf{F}$ with $\lambda \neq 0$ then $(\lambda A)^{\circ} = |\lambda|^{-1} A^{\circ}$. |
(4) | $\displaystyle{\left ( \bigcup_{\alpha} A_{\alpha} \right )^{\circ} = \bigcap_{\alpha} A_{\alpha}^{\circ}}$. |
- On The Polar of a Subspace page we proved that if $(E, F)$ is a dual pair and $M$ is a subspace of $E$ then the polar $M^{\circ}$ in $F$ consists of all points $y \in F$ such that $\langle x, y \rangle = 0$ for all $x \in M$.
- In particular for the dual pair $(E, E^*)$, if $M$ is a subspace of $E$ then the polar $M^{\circ}$ in $E^*$ is $M^{\perp}$.
- On the The Polar Criterion for Equicontinuity of a Set of Linear Forms page, for Hausdorff locally convex topological vector spaces we classified the notion of equicontinuity in terms of polar sets:
If $E$ is a Hausdorff locally convex topological vector space so that $(E, E')$ is a dual pair, then a subcollection $\mathcal V$ of continuous linear forms on $E$ is equicontinuous if and only if there exists a neighbourhood $U$ of the origin for which $\mathcal V \subseteq U^{\circ}$.
- On The Bipolar of a Set page said that if $(E, F)$ and $(F, G)$ are dual pairs and $A \subseteq E$ then the Bipolar of $A$ (in $G$) denoted by $A^{\circ \circ}$ is defined to be the polar (in $G$) of the polar of $A$ (in $F$).
- If $(E, F)$ and $(F, G)$ are dual pairs for which $E \subseteq G \subseteq F^*$ (as subspaces) then many useful properties can be deduced. These properties can be found on the following pages:
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair, A ⊆ A°°
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair then A°° is the σ(G, F)-Closed Absolutely Convex Hull of A
- If E is a Hausdorff LCTVS then A°° (in E) is the Closed Absolutely Convex Hull of A
- These results are summarized below.
Results Concerning Bipolars of Sets | |
Characterization of Points in the Bipolar | If $A \subseteq E$ then $z \in A^{\circ \circ}$ if and only if $|\langle z, y \rangle| \leq \sup \{ |\langle x, y \rangle| : x \in A \}$ for all $y \in F$. |
$A$ is Contained in its Bipolar | if $A \subseteq E$ then $A \subseteq A^{\circ \circ}$. |
Classification of Bipolar of $A$ | If $A \subseteq E$ then $A^{\circ \circ}$ is the $\sigma(G, F)$-closed absolutely convex hull of $A$, i.e, $\displaystyle{A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}^{\sigma(G, F)}}$. |
Classification of the Bipolar of $A$ (in $E$) | If $E$ is a Hausdorff locally convex topological vector space (so that $(E, E')$ and $(E', E)$ are dual pairs replacing $(E, F)$ and $(F, G)$ above) and if $A \subseteq E$ then the bipolar $A^{\circ \circ}$ (in $E$) is the closed absolutely convex hull of $A$, i.e., $A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}$. |
- Lastly, on The Polar of an Intersection of σ(E, F)-Closed Absolutely Convex Sets page we proved that if $\{ A_{\alpha} \}_{\alpha}$ is a collection of $\sigma(E, F)$-closed absolutely convex subsets of $E$ then:
\begin{align} \quad \left ( \bigcap_{\alpha} A_{\alpha} \right )^{\circ} = \overline{\mathrm{abs \: conv}\left ( \bigcup_{\alpha} A_{\alpha}^{\circ} \right )}^{\sigma(F, E)} \end{align}