# 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.

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

## 1 Answer

The Schwartz space (of infinitely differentiable functions) is just an example of a Schwartz space. The term “Schwartz space” is used for two concepts. The German Wikipedia has two articles: Schwartz-Raum and Schwartz-Raum (allgemein), which includes a bibliography.

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The German wikis are much more comprehensive than the English one (though I can't read German, I used google translator). Anyway, I'll have look at those references and come back soon. – James Jun 25 '13 at 10:29
“allgemein” means “general”, referring to the broader concept (and “Raum” means “space”). – The User Jun 25 '13 at 10:33
The German wiki Schwart-Raum (allgemein) and the included bibliography (especially that of Jarchow locally convex spaces) were extremely helpful. Thank you very much. Someone perhaps should add an entry for the general concept of Schwartz spaces in the English Wikipedia to avoid any confusion (I don't think I qualify to do so) – James Jun 25 '13 at 11:44