Definition:Locally Convex Space

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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:

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


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

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

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

Also see

  • Results about locally convex spaces can be found here.