Review of Dual Pairs
Review of Dual Pairs
- On the Dual Pairs of Vector Spaces page we defined a Dual Pair to be a triple $(E, F, \langle \cdot, \cdot \rangle)$ where $E$ and $F$ are vector spaces and $\langle \cdot, \cdot \rangle : E \times F \to \mathbf{F}$ is a bilinear form with the property that for each nonzero $x \in E$ there exists a $y \in F$ such that $\langle x, y \rangle \neq 0$ and with the property that for each nonzero $y \in F$ there exists a $x \in E$ such that $\langle x, y \rangle \neq 0$.
- We then observed some common examples of dual pairs:
Examples of Dual Pairs | |
Example 1: | If $E$ is a vector space then $(E, E^*)$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E^*$ by $\langle x, f \rangle := f(x)$. |
Example 2: | If $E$ is a Hausdorff locally convex topological vector space then $(E, E')$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E'$ by $\langle x, f \rangle := f(x)$. |
Example 3: | If $(E, F, \langle \cdot, \cdot \rangle)$ is a dual pair then $(F, E, [\cdot, \cdot])$ with the duality pairing given for all $x \in E$ and for all $y \in F$ by $[y, x] := \langle x, y \rangle$. |
- On The Weak Topology on E Determined by F page, for a dual pair $(E, F)$, we defined the Weak Topology on $E$ Determined by $F$ denoted by $\sigma(E, F)$, to be the coarsest topology determined by the collection of seminorms $Q := \{ p_y : y \in F \}$ where for each $y \in F$ we defined $p_y : E \to \mathbf{F}$ by:
\begin{align} \quad p_y(x) := |\langle x, y \rangle| \end{align}
- We also noted the following properties of the weak topology on $E$ determined by $F$, which are summarized below:
Properties of the Weak Topology on $E$ Determined by $F$ | |
(1) | $\sigma(E, F)$ is the coarsest topology for which each of the seminorms $p_y$ are continuous. |
(2) | $\sigma(E, F)$ is a Hausdorff locally convex topology. |
(3) | A base of $\sigma(E, F)$-closed neighbourhoods of the origin are given by sets of the form $\displaystyle{\{ x : \sup_{1 \leq i \leq n} p_{y_i}(x) \leq 1 \}}$ with $y_1, y_2, ..., y_n \in F$. |
- On the The Topological Dual of E Equipped with σ(E, F) is F page we classified the topological dual of $E$ equipped with the $\sigma(E, F)$ topology:
If $(E, F)$ is a dual pair then the topological dual of $E$ equipped with $\sigma(E, F)$ is $F$, i.e.:
$(E^{\sigma(E, F)})' = F$
- On the Topologies of the Dual Pair (E, F) page we said that if $(E, F)$ is a dual pair then a Topology of the Dual Pair $(E, F)$ is a topology $\tau$ on $E$ that is locally convex with the property that $(E^{\tau})' = F$. Thus, the weak topology on $E$ determined by $F$ is a topology of the dual pair.
- We then proved a very important result regarding topologies of the dual pair $(E, F)$ and convex sets:
If $(E, F)$ is a dual pair and $A$ is a convex subset of $E$ then the closure of $A$ is the same subset when $E$ is equipped with any topology of the dual pair $(E, F)$.