Definition:Locally Convex Space

Definition

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

Let $X$ be a vector space over $\GF$.

Let $\mathcal P$ be a set of seminorms on $X$.

We call the pair $\struct {X, \mathcal P}$ a locally convex space over $\GF$.

Hausdorff Locally Convex Space

We say that $\struct {X, \mathcal P}$ is Hausdorff if and only if $\PP$ is separating.

Standard Topology

For each $p \in \PP$, $\epsilon > 0$ and $y \in X$, define:

$\map {B_p} {\epsilon, x} = \set {y \in X : \map p {y - x} < \epsilon}$

and:

$\SS = \set {\map {B_p} {\epsilon, x} : p \in \PP, \, \epsilon > 0, \, x \in X}$

Let $\tau$ be the topology generated by $\SS$.

We call $\tau$ the standard topology on $\struct {X, \mathcal P}$.

Also see

• Characterization of Locally Convex Spaces
• Locally Convex Space is Topological Vector Space shows that locally convex spaces with their standard topology are topological vector spaces, and so theorems applicable to topological vector spaces may be used.

• Results about locally convex spaces can be found here.