I have a very general question regarding the book "Kriegl, Michor: The Convenient Setting of Global Analysis.": Is the differential calculus of locally convex spaces (see here, for instance) canonically embedded in the differential calculus wrt the $c^\infty$-topology (Dfn 3.11.)? More precisely, for two lcs $X,Y$, does the inclusion $C^\infty_{lcs}(X,Y) \subseteq C^\infty_{c^\infty}(X,Y)$ hold?
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$\begingroup$ So, does the question make no sense or is the answer just not known? As far as i understood, this convenient setting is a generalization of locally convex calculus in order to obtain nice properties such as the exponential law, right? $\endgroup$– Bipolar MindsCommented Mar 24, 2015 at 10:49
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$\begingroup$ In fact, the reference you mention answers precisely this question on the last page... $\endgroup$– Stefan WaldmannCommented May 6, 2015 at 7:17
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It is actually well known that every smooth map in the Michal-Bastiani (MB) sense (what you call "differential calculus on locally convex spaces") is also smooth in the convenient sense.
For a quick proof, just note that in both calculi the definitions of a smooth curve from the reals into a locally convex vector space coincide. To see that an (MB) smooth map $f \colon E \supseteq U \rightarrow F$ is convenient smooth, consider a smooth curve $c \colon \mathbb{R} \rightarrow U$. Now the chain rule (for the (MB)-calculus) implies that $f \circ c$ is a smooth curve. Hence we deduce that $f$ takes smooth curves to smooth curves and is thus convenient smooth. The converse is of course false in general (but holds for example on complete metric spaces).
To sum up, there is always a canonical inclusion (of sets) of the MB-smooth maps into convenient smooth maps.
answered May 6, 2015 at 6:34
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$\begingroup$ @Stefan Waldmann: I should have checked the whole reference given, not just the definition of the calculus... $\endgroup$ Commented May 6, 2015 at 7:52
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1$\begingroup$ "there is always a canonical inclusion (of sets) of the MB-smooth maps into convenient smooth maps." For this to hold, one should add the requirement that one restricts to considering only MB-smooth maps $f:E\supseteq U\to F$ where $E,F$ are "convenient" spaces. $\endgroup$– TaQCommented Feb 21, 2016 at 10:47
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$\begingroup$ @TaQ: thats true thanks for the added accuracy $\endgroup$ Commented Apr 15, 2016 at 6:59