A characteristic property of the space s

TL;DR

The paper characterizes the space s of rapidly decreasing sequences by a stability condition and shows its isomorphism with certain restriction spaces of smooth functions, linking sequence spaces and function spaces.

Contribution

It introduces a stability condition that characterizes the space s and applies this to identify restriction spaces of smooth functions with s.

Findings

Complemented subspaces of s are isomorphic to s under certain conditions.

The space of restrictions of smooth functions on a Cantor set is isomorphic to s.

The results extend the theory of restriction spaces of smooth functions.

Abstract

It is shown that under certain stability conditions a complemented subspace of the space $s$ of rapidly decreasing sequences is isomorphic to $s$ and this condition characterizes $s$. This result is used to show that for the classical Cantor set $X$ the space $C_\infty(X)$ of restrictions to $X$ of $C^\infty$-functions on $\R$ is isomorphic to $s$, so completing the theory developed in "Restriction spaces of $A^\infty$", to appear in Rev. Mat. Iberoamericana 29.4 (2013)