The Weak Topology on E Determined by F

# The Weak Topology on E Determined by F

Let $(E, F, \langle \cdot, \cdot \rangle)$ be a dual pair. For each $y \in F$, observe that the function $p_y : E \to [0, \infty)$ defined by:

(1)
\begin{align} p_y(x) := | \langle x, y \rangle | \end{align}

is a seminorm on $E$. Thus we can consider the coarsest topology on $E$ determined by the family of seminorms $\{ p_y : y \in F \}$.

 Definition: Let $(E, F, \langle \cdot, \cdot \rangle)$ be a dual pair. The Weak Topology on $E$ Determined by $F$, denoted by $\sigma (E, F)$, is the coarsest topology determined by the set of seminorms $\{ p_y : y \in F \}$.

We will state some important properties of $\sigma (E, F)$.

 Proposition 1: Let $(E, F)$ be a dual pair. Then: (1) $E$ equipped with $\sigma(E, F)$ is the coarsest topology for which each $p_y$ is $\sigma(E, F)$-continuous. (2) $E$ equipped with $\sigma(E, F)$ is a locally convex topological vector space. (3) $E$ equipped with $\sigma(E, F)$ is Hausdorff. (4) A base of closed $\sigma(E, F)$-neighbourhoods of the origin are given by sets of the form $\displaystyle{\{ x \in E : \sup_{1 \leq i \leq n} p_{y_i}(x) \leq 1 \}}$ with $y_1, y_2, ..., y_n \in F$.
• Proof: (1), (2), and (4) are immediate by the definition of $\sigma(E, F)$ being the topology determined by $\{ p_y : y \in F \}$. For (3), observe that since $E$ equipped with $\sigma(E, F)$ is a locally convex Hausdorff space and since for each nonzero $x \in E$ we have that there exists a $y \in F$ such that $p_y(x) = |\langle x, y \rangle| \neq 0$, we conclude by the proposition on the Criterion for the Coarsest Topology Determined by a Set of Seminorms to be Hausdorff page that $E$ equipped with $\sigma(E, F)$ is Hausdorff. $\blacksquare$