Definition:Weak Topology on Topological Vector Space

Definition
Let $K$ be a topological field.

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

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

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

We say that $w$ is the weak topology on $X$ :


 * for each $x \in X \setminus \set {\mathbf 0_X}$ there exists $f \in X^\ast$ such that $\map f x \ne 0$.

That is, $w$ "separates the points of $X$".

Also see

 * Initial Topology on Hausdorff Locally Convex Space is Weak Topology
 * Initial Topology on Normed Vector Space is Weak Topology