Definition:Locally Convex Space/Standard Topology

Definition

Let $\struct {X, \mathcal P}$ be a locally convex space.

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 $\struct {X, \mathcal P}$.

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $2.1$: Normed Spaces