Schwartz Space - Definition

Definition

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

where α, β are multi-indices, C∞(Rn) is the set of smooth functions from Rn to C, and

Here, sup denotes the supremum, and we again use multi-index notation.

To put common language to this definition, we could note that a rapidly decreasing function is 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 inverse power of x. Especially, S(Rn) is a subspace of the function space C∞(Rn) of infinitely smooth functions.