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 :


 * 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.