nLab
Schwartz space

Contents

Idea

In functional analysis,

  1. a Schwartz space (Terzioglu 69, Kriegl-Michor 97, below 52.24) is a locally convex topological vector space EE with the property that whenever UU is an absolutely convex neighbourhood of 00 then it contains another, say VV, such that UU maps to a precompact set in the normed vector space E VE_V.

  2. the Schwartz space of an open subset of Euclidean space is the space of functions with rapidly decreasing partial derivatives (def. below). On this space the operation of Fourier transform is a linear automorphism (prop. below). The continuous linear functionals on this space are the tempered distributions.

The Schwartz spaces in the second sense are examples of the Schwartz spaces in the first sense.

Definition

Definition

(functions with rapidly decreasing partial derivatives)

For nn \in \mathbb{N}, a smooth function f: nf \colon \mathbb{R}^n \to \mathbb{R} on the Euclidean space n\mathbb{R}^n has rapidly decreasing partial derivatives if the absolute value of the product of any partial derivative βf\partial_\beta f of the function with any polynomial function is a bounded function:

α,β n(supx nx α βf(x)<K α,β) \underset{\alpha, \beta \in \mathbb{N}^n}{\forall} \left( \underset{x \in \mathbb{R}^n}{sup} {\Vert x^\alpha \partial_{\beta} f(x) \Vert} \lt K_{\alpha, \beta} \right)

for some choices of positive constants K α,βK_{\alpha, \beta}.

(e.g. Hörmander 90, def. 7.1.2)

Definition

(Schwartz space of functions with rapidly decreasing partial derivatives)

For nn \in \mathbb{N} the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n) is the topological vector space whose

(e.g. Hörmander 90, def. 7.1.2)

Proposition

(the Schwartz space is a Fréchet space)

The Schwartz space 𝒮\mathcal{S} of functions with rapidly decreasing partial derivatives (def. ) is a Fréchet space.

(e.g. p. 2 here: pdf)

Definition

(tempered distributions)

A tempered distribution on n\mathbb{R}^n is a continuous linear functional on the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n) (def. ).

(e.g. Hörmander 90, def. 7.1.7)

Properties

Proposition

(Fourier transform is linear automorphism of Schwartz space)

For nn \in \mathbb{N} the operation of Fourier transform ff^f \mapsto \hat f is well defined on all smooth functions on n\mathbb{R}^n with rapidly decreasing derivatives (def. ) and indeed constitutes a linear isomorphism from the Schwartz space (def. ) to itself:

()^:𝒮( n)𝒮( n) \widehat {(-)} \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathbb{R}^n)

(e.g. Hörmander 90, lemma 7.1.3, Melrose 03, theorem 1.3)

References

Named by Alexander Grothendieck after Laurent Schwartz (according to Terzioglu 69).

The general concept of Schwartz spaces appears in

Specifically the Schwartz spaces of functions with rapidly decreasing partial derivatives are discussed for instance in

  • Lars Hörmander, section 7.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

  • Richard Melrose, chapter 1 of Introduction to microlocal analysis, 2003 (pdf)

See also

Last revised on February 16, 2018 at 14:01:46. See the history of this page for a list of all contributions to it.