TITLE: What is the definition of being smooth for a function from a Lie group to a Fréchet space?
QUESTION [5 upvotes]: In representation theory of real groups, one is confronted with the notion of smoothness for functions defined on a Lie group with values in a Fréchet space (e.g. see Wallach's Real Reductive Groups I, 1.1.1). What is the precise definition of this? Presumably, the definition should be given in terms of local coordinates, so I am actually asking what smoothness means for a function defined on an open subset of $\mathbb{R}^n$ with values in a Fréchet space $V$. (If $V$ is a Banach space or $n = 1,$ I have no problem.) Proceeding in the usual manner, at some point one has to see $\mathrm{Hom}(\mathbb{R}^n,V)$ as a topological vector space of the same kind as $V$ in order to be able to define the second derivative and so on. On p. 52 of his Representation Theory of Semisimple Groups, Knapp mentions that there exists a canonical topological vector space structure on $\mathrm{Hom}(\mathbb{R}^n,V).$ What is this canonical structure? Does it turn $\mathrm{Hom}(\mathbb{R}^n,V)$ into a topological vector space of the same kind as $V$?

REPLY [3 votes]: Since the above answers and comments refer to the concept of differentiability for functions not just with values IN an infinitely dimensional space, but also defined ON one, despite the fact that your question addresses functions defined on a Lie group and so a finite dimensional space and you explicitly state that this reduce to the case of functions on euclidean space by localisation, let me take the liberty of adding some information on this situation.  This was considered in some detail in the article "Espaces de fonctions différentiables a valeurs vectorielles", Jour. d'Anal. 4 (1954-56) 88-148 by Laurent Schwartz  which is probably still the definitive statement on the subject.  The definition is exactly as in the scalar case using limits of difference quotients.  Schwartz uses the theory of nuclear spaces and tensor products which had just been invented by 
Grothendieck for the somewhat analogous case of holomorphic functions and the fact that the smooth functions form a nuclear space makes the theory somewhat simpler.