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](http://www.mathematik.tu-darmstadt.de/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2005/Liegruppen/skript/ch2vorl.pdf), 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?



Relation between locally convex calculus and Kriegl & Michor's "convenient setting"
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. 


