---
tag: 87FM669CDD
kind: definition
name: Nuclear space
msc: [46A11]
defines: XC6XPMC2FQ
uses: [B6GY2KJMWW, FP5GVCN8GV, XPANM7C1D3]
sources:
  - url: https://en.wikipedia.org/wiki/Nuclear_space
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $E$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\mathbb{K}$.
For a continuous [[seminorm|FP5GVCN8GV]] $p$ on $E$, let $E_p$ be the Banach space obtained by completing the quotient $E / \set{x : p(x) = 0}$ with the norm induced by $p$.
If $q \ge p$ are continuous seminorms, the identity of $E$ induces a continuous linear map $\iota_{q, p} \colon E_q \to E_p$ of norm at most $1$.
$E$ is *nuclear* if the following holds.
For every continuous seminorm $p$ on $E$ there is a continuous seminorm $q \ge p$ such that $\iota_{q, p}$ is a [[nuclear operator|XPANM7C1D3]].

## Remarks

Example: the Schwartz space $\mathcal{S}(\R^n)$, the space $C^\infty(\Omega)$ of smooth functions on an open set $\Omega \subseteq \R^n$, and the space $H(\Omega)$ of holomorphic functions on an open set $\Omega \subseteq \C$ are nuclear.
Non-example: an infinite-dimensional normed space $E$ is not nuclear.
Take $p$ to be the norm; then every continuous seminorm $q \ge p$ is equivalent to the norm.
So $\iota_{q, p}$ is an isomorphism of infinite-dimensional Banach spaces, which is not compact and hence not nuclear.
Grothendieck's original definition is equivalent: $E$ is nuclear if and only if, for every Banach space $F$, the projective and injective tensor topologies on $E \otimes F$ agree.
Every nuclear Fréchet space is a Montel space.