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 , particulary, 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 .