---
tag: B6GY2KJMWW
kind: concept
name: Locally convex space
msc: [46A03]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space
same_as:
  wikidata: Q1572357
  wikipedia: Locally convex topological vector space
---

## Summary

A locally convex space is a topological vector space in which every neighbourhood of $0$ contains a convex neighbourhood of $0$.
Equivalently, its topology is [[generated by a family of seminorms|ZVFXQ7S6FY]].
Normed spaces, [[Fréchet spaces|3GAA614673]], spaces of smooth functions and spaces of distributions are all locally convex.
Local convexity is what makes the Hahn–Banach theorem work, so these spaces have many continuous linear forms and a rich duality theory.

## Explanations

### L1 human

In a normed space the small balls around $0$ are convex: the segment between two points of a ball stays in the ball.
A locally convex space keeps this feature, but the small neighbourhoods need not come from one norm.
They may come from many seminorms used together.
For example, the space of all real sequences, with convergence meaning convergence of each coordinate, is locally convex but has no norm giving that convergence.
Some natural spaces fail local convexity, such as $L^p([0, 1])$ for $0 < p < 1$.