In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Schwartz space
is the
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
of all
functions whose
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s 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.

Schwartz space is named after French mathematician
Laurent Schwartz.
Definition
Let
be the
set of non-negative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, 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 such that , , , ... all exist everywhere on and go to zero as faster than any reciprocal power of . In particular, is a
subspace of the function space (, ) of smooth functions from into .
Examples of functions in the Schwartz space
* If
is a multi-index, and ''a'' is a positive
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, then
*:
* Any smooth function ''f'' with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed ...
is in . This is clear since any derivative of ''f'' is
continuous and supported in the support of ''f'', so (
has a maximum in R
''n'' by the
extreme value theorem.
* Because the Schwartz space is a vector space, any polynomial
can be multiplied by a factor
for
a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials into a Schwartz space.
Properties
Analytic properties
* From
Leibniz's rule, it follows that is also closed under
pointwise multiplication:
*: If then the product .
In particular, this implies that is an -algebra. More generally, if and is a bounded smooth function with bounded derivatives of all orders, then .
* The Fourier transform is a
linear isomorphism .
* If then is
Lipschitz continuous and hence
uniformly continuous on .
* is a
distinguished locally convex Fréchet Schwartz TVS over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s.
* Both ''and'' its
strong dual space are also:
#
complete Hausdorff locally convex spaces,
#
nuclear Montel spaces,
#
ultrabornological spaces,
#
reflexive barrelled Mackey spaces.
Relation of Schwartz spaces with other topological vector spaces
*If , then .
*If , then is
dense in .
*The space of all
bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
s, , is included in .
See also
*
Bump function
In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
*
Schwartz–Bruhat function
*
Nuclear space
References
Sources
*
*
*
*
{{Functional analysis
Topological vector spaces
Smooth functions
Fourier analysis
Function spaces
Schwartz distributions