In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function.

A two-dimensional Gaussian function is an example of a rapidly decreasing function.

Schwartz space is named after French mathematician Laurent Schwartz.


Let be the set of non-negative integers, and for any , let be the n-fold Cartesian product.

The Schwartz space or space of rapidly decreasing functions on is the function space

where is the function space of smooth functions from into , and
Here, denotes the supremum, and we used multi-index notation, i.e. and .

To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f′(x), f′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x. In particular, S(Rn, C) is a subspace of the function space C(Rn, C) of smooth functions from Rn into C.

Examples of functions in the Schwartz space


Analytic properties

In particular, this implies that 𝒮(Rn) is an R-algebra. More generally, if f ∈ 𝒮(R) and H is a bounded smooth function with bounded derivatives of all orders, then fH ∈ 𝒮(R).

  1. complete Hausdorff locally convex spaces,
  2. nuclear Montel spaces,
It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology,[1]
  1. Ultrabornological spaces,
  2. reflexive barrelled Mackey spaces.

Relation of Schwartz spaces with other topological vector spaces

See also


  1. ^ Trèves 2006, pp. 351–359.


This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.