---
tag: 3GAA614673
kind: concept
name: Fréchet space
msc: [46A04]
specializes: [B6GY2KJMWW, 86BYEV9AX8]
sources:
  - url: https://en.wikipedia.org/wiki/Fréchet_space
same_as:
  wikidata: Q1471397
  wikipedia: Fréchet space
---

## Summary

A Fréchet space is a [[locally convex space|B6GY2KJMWW]] that is also an [[F-space|86BYEV9AX8]]: its topology comes from a complete translation-invariant metric.
Equivalently, it is a complete Hausdorff space whose topology is generated by a countable family of seminorms.
Fréchet spaces generalize Banach spaces and include the main spaces of smooth and holomorphic functions.
The Hahn–Banach theorem, the open mapping theorem and the uniform boundedness principle all hold for them.

## Explanations

### L1 human

A Banach space is complete for one norm.
A Fréchet space is complete for a whole sequence of seminorms used together.
The space of smooth functions on $\R$ is a typical case: one controls $f$, $f'$, $f''$, and so on, each on bounded intervals.
No single norm captures all this control, but countably many seminorms do.
Convergence means convergence for each seminorm separately.