---
tag: T32W553W95
kind: concept
name: Topological vector space
msc: [46A03, 57N17]
specializes: [000AYPA9AJ, 92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_vector_space
same_as:
  wikidata: Q1455249
  wikipedia: Topological vector space
---

## Summary

A topological vector space is a [[vector space|000AYPA9AJ]] over $\R$ or $\C$ with a topology in which addition and scalar multiplication are continuous.
It is the common frame of functional analysis: normed spaces, Banach spaces, Hilbert spaces and locally convex spaces are all examples.
Under addition it is a commutative [[topological group|92700QAB5Y]], so its topology is determined by the neighbourhoods of $0$.
Continuity of linear maps, convergence of series and duality are all studied in this setting.

## Explanations

### L1 human

A topological vector space is a space of vectors where you can also say when vectors are close to each other.
Adding vectors and stretching them by numbers must respect this closeness.
The plane $\R^2$ with its usual distance is the simplest example.
Spaces of functions, such as all continuous functions on $[0, 1]$ with the maximum distance, are the examples that matter most in analysis.