---
tag: GE020MY43M
kind: definition
name: Schwartz space
msc: [46F05, "46E10"]
defines: CG9JEATW3Q
uses: [0V33BNKZ9S, 0PJXBR6QA3, 3GAA614673]
sources:
  - url: https://en.wikipedia.org/wiki/Schwartz_space
---

## Definition

The *Schwartz space* $\mathcal{S}(\R^n)$ is the set of [[smooth functions|0V33BNKZ9S]] $f \colon \R^n \to \C$ such that
$$p_{\alpha, \beta}(f) = \sup_{x \in \R^n} \abs{x^\alpha \, \partial^\beta f(x)} < \infty$$
for all [[multi-indices|0PJXBR6QA3]] $\alpha, \beta \in \N^n$.
Its elements are called *Schwartz functions* or *rapidly decreasing functions*.
With the family of seminorms $p_{\alpha, \beta}$, the space $\mathcal{S}(\R^n)$ is a [[Fréchet space|3GAA614673]].

## Remarks

Example: the Gaussian $f(x) = e^{-\abs{x}^2}$ is a Schwartz function.
Each derivative $\partial^\beta f$ is a polynomial times $e^{-\abs{x}^2}$, and a polynomial times $e^{-\abs{x}^2}$ is bounded on $\R^n$.
Every [[test function|HS65DW0HXH]] on $\R^n$ is a Schwartz function.
Non-example: $f(x) = 1/(1 + x^2)$ on $\R$ is smooth, but $p_{2, 0}(f) = \sup_x x^2/(1 + x^2) = 1$ while $p_{3, 0}(f) = \sup_x \abs{x}^3/(1 + x^2) = \infty$.
The space is closed under differentiation, under multiplication by polynomials and under the [[Fourier transform|DPVM34ADZV]], which is a bijection of $\mathcal{S}(\R^n)$ onto itself.
Equivalently, $f$ is a Schwartz function if and only if $(1 + \abs{x})^N \abs{\partial^\beta f(x)}$ is bounded for every $N \in \N$ and every $\beta$.

## Explanations

### L1 human

A Schwartz function is a smooth function that, with all its derivatives, decays faster than every inverse power of $\abs{x}$.