# Relation between locally convex calculus and Kriegl & Michor's “convenient setting”

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?

• 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? – Bipolar Minds Mar 24 '15 at 10:49
• In fact, the reference you mention answers precisely this question on the last page... – Stefan Waldmann May 6 '15 at 7:17

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).
• "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. – TaQ Feb 21 '16 at 10:47