Definition:Weak-* Topology

Theorem
Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $X^\ast$ be the topological dual space of $X$.

For each $x \in X$, define $x^\wedge : X^\ast \to \Bbb F$ by:


 * $\map {x^\wedge} f = \map f x$

Let:


 * $\sigma = \set {x^\wedge : x \in X}$

Let $w^\ast$ be the initial topology on $X^\ast$ with respect to $\sigma$.

We say that $w^\ast$ is the weak-$\ast$ topology of $X^\ast$.