# Schwartz space defined on locally convex spaces

This is my first post here, so bear with me ;)

In wikipedia and other references, Schwartz space is defined as the set of inﬁnitely differentiable functions on $\mathbb{R}^n$. On the other hand, A locally convex space X is named a LS-space if there is a Fréchet-Schwartz space Y such that the strong dual $Y^{'}$ is isomorphic to X. (Naturally, A Fréchet-Schwartz space is a Fréchet space which is at the same time a Schwartz space). It seems to me that Schwartz spaces in the definition of Fréchet-Schwartz spaces and LS-spaces are considerably different than the definition given in wikipedia. After all they have to be defined in the context of locally convex spaces. I appreciate it if someone could clarify the definition of Schwartz space here.

-
That is not what Wikipedia says, at least at this minute. –  Yemon Choi Jun 25 '13 at 10:11