Schwartz space
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.

Schwartz space is named after French mathematician Laurent Schwartz.

Definition

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 spacewhere  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 α is a multi-index, and a is a positive real number, then

 Any smooth function f with compact support is in S(Rn). This is clear since any derivative of f is continuous and supported in the support of f, so (xαDβ) f has a maximum in Rn by the extreme value theorem.
 Because the Schwartz space is a vector space, any polynomial  can by multiplied by a factor  for  a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.

Properties

Analytic properties 

 From Leibniz's rule, it follows that  is also closed under pointwise multiplication:
 If  then the product .
 The Fourier transform is a linear isomorphism .
 If  then  is uniformly continuous on .
 is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers.
 Both  and its strong dual space are also:
complete Hausdorff locally convex spaces,
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,
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 functions, , is included in .

See also
 Bump function
 Schwartz–Bruhat function
 Nuclear space

References

Sources

 

Topological vector spaces
Smooth functions
Fourier analysis
Function spaces
Schwartz distributions