---
tag: N8TGDDJCFJ
kind: definition
name: Fréchet space
msc: [46A04]
defines: 3GAA614673
uses: [B6GY2KJMWW, 86BYEV9AX8]
sources:
  - url: https://en.wikipedia.org/wiki/Fréchet_space
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *Fréchet space* over $\mathbb{K}$ is a topological vector space $E$ over $\mathbb{K}$ that is both [[locally convex|B6GY2KJMWW]] and an [[F-space|86BYEV9AX8]].

## Remarks

Equivalently, the topology of $E$ is generated by a countable family of [[seminorms|FP5GVCN8GV]] $p_1 \le p_2 \le \cdots$ that separates points, and $E$ is complete.
From such seminorms one gets a translation-invariant metric
$$d(x, y) = \sum_{n = 1}^{\infty} 2^{-n} \frac{p_n(x - y)}{1 + p_n(x - y)}$$
that induces the topology.
Example: the space $C(\R)$ of continuous real functions with the seminorms $p_n(f) = \sup_{\abs{t} \le n} \abs{f(t)}$ is a Fréchet space.
A sequence that is Cauchy for all $p_n$ converges uniformly on each $[-n, n]$, and the limit is continuous.
Every Banach space is a Fréchet space.
Non-example: $L^p([0, 1])$ with $0 < p < 1$ is an F-space but is not locally convex, so it is not a Fréchet space.
Some older texts use the name Fréchet space for what is here called an F-space.