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 $X$ if and only if:

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, if and only if $w$ "separates the points of $X$".

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Also see

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

• Results about weak topologies on topological vector spaces can be found here.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $3.11$: The weak topology of a topological vector space