Definition:Locally Convex Topological Vector Space

Definition

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a topological vector space over $\GF$.

We say that $\struct {X, \tau}$ is a locally convex topological vector space if and only if:

there exits a local basis $\BB$ for ${\mathbf 0}_X$ in $\struct {X, \tau}$ such that:

each $A \in \BB$ is convex.

Also see

• Characterization of Locally Convex Topological Vector Space shows that the locally convex topological vector spaces are precisely the locally convex spaces equipped with their standard topology.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.8$: Types of topological vector spaces