Schwartz space

In mathematics, Schwartz space formula_1 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 formula_2 of formula_1, 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.
Motivation.
The idea behind the Schwartz space is to consider the set of all smooth functions on formula_4 which decrease rapidly. This is encoded by considering all possible derivatives formula_5 (with multi-index formula_6) on a smooth complex-valued function formula_7 and the supremum of all possible values of formula_8 multiplied by any monomial formula_9 and bounding them. This restriction is encoded as the inequalityformula_10Notice if we only required the derivatives to be bounded, i.e., formula_11this would imply all possible derivatives of a smooth function must be bounded by some constant formula_12, soformula_13For example, the smooth complex-valued function formula_14 with formula_15 gives formula_16, which is an unbounded function, so any polynomial could not be in this space. But, if we require in addition the original inequality, then this result is even stronger because it implies the inequalityformula_17 for any formula_18 and some constant formula_19 since formula_20This demonstrates the growth of all derivatives of formula_21 must be "far" lesser than the inverse of any monomial.
Definition.
Let formula_22 be the set of non-negative integers, and for any formula_23, let formula_24 be the "n"-fold Cartesian product. The "Schwartz space" or space of rapidly decreasing functions on formula_4 is the function spaceformula_26where formula_27 is the function space of smooth functions from formula_4 into formula_29, andformula_30Here, formula_31 denotes the supremum, and we use multi-index notation.
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 .