---
tag: BW9BBF244W
kind: definition
name: Locally convex space
msc: [46A03]
defines: B6GY2KJMWW
uses: [02AS45T3H9, 0ZCY0Z80B6, ZVFXQ7S6FY]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a topological vector space over $\mathbb{K}$.
$E$ is *locally convex* if every [[neighbourhood|0ZCY0Z80B6]] of $0$ in $E$ contains a [[convex|02AS45T3H9]] neighbourhood of $0$.

## Remarks

Equivalently, the topology of $E$ is the [[topology generated by a family of seminorms|ZVFXQ7S6FY]], namely the family of all continuous seminorms on $E$.
Many authors also require $E$ to be Hausdorff.
Example: every normed space is locally convex, since the open balls $\set{x : \norm{x} < r}$ are convex and form a neighbourhood base at $0$.
Convexity of a ball: if $\norm{x}, \norm{y} < r$ and $0 \le t \le 1$, then $\norm{t x + (1 - t) y} \le t \norm{x} + (1 - t) \norm{y} < r$.
Non-example: for $0 < p < 1$, the space $L^p([0, 1])$ with the metric $d(f, g) = \int_0^1 \abs{f - g}^p$ is a topological vector space.
But its only convex open set containing $0$ is the whole space, so it is not locally convex.