Schwartz room

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The Schwartz space is a function space that is examined in the mathematical sub-area of functional analysis. This is named after the mathematician Laurent Schwartz , who provided central results in distribution theory , whereby the Schwartz space also played an important role. The elements of the Schwartz space are called Schwartz functions . A special feature of this space is that the Fourier transform forms a linear automorphism in this space.


A function is called a Schwartz function or rapidly decreasing if it is continuously differentiable any number of times , and if the function is restricted to for all multi-indices , where the -th derivative denotes.

The vector space of all Schwartz functions is called Schwartz space and is denoted by. In a nutshell, then

The Schwartz space is a locally convex space that can be metrised , which is defined by the family of semi-norms

is induced.


  • The functions are based on Schwartz functions .
  • Every function that can be differentiated as often as required with a compact carrier is a Schwartz function. The vector space of the test functions with compact support is therefore a real subspace of the Schwartz space.
  • The Hermitian functions are also Schwartz functions.


  • The Schwartz space is complete in terms of the topology (or metric) induced by the family of semi-norms , and is thus a Fréchet space . He also has the Montel property .
  • The Fourier transform forms a linear automorphism on Schwartz space.
  • As mentioned in the examples, the space of smooth functions with compact support is a subspace of the Schwartz space. This is even close to the Schwartz room.
  • The Schwartz room is separable .
  • for .
  • For the Schwartz space lies close to the space of the p-integrable functions
  • With the help of this density argument one can define the Fourier transformation on the Hilbert space .

Tempered distributions

A continuous , linear mapping is called temperature-controlled distribution . The set of all tempered distributions is denoted by. This is the topological dual space too .


  • Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).

Individual evidence

  1. a b Man Wah Wong: An introduction to pseudo-differential operator . World Scientific, River Edge, NJ 1999, ISBN 978-981-02-3813-1 , pp. 10-11 .