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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 $V$?

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There is a book by A.Kriegl and P.Michor, "The Convenient Setting of Global Analysis"… They consider questions like yours, but in general sutiation. – Sergei Akbarov Sep 8 '13 at 5:09

For two locally convex topological vector spaces $X,Y$, there are at least two useful topologies on ${\rm Hom}(X,Y)$. The stronger one is given by seminorms $\nu_{x,\mu}(\phi)=\mu(\phi(x))$ as $x$ ranges over $X$ and $\mu$ ranges over seminorms giving the topology on $Y$, and the weaker is given by seminorms $\nu_{x,\lambda}(\phi)=|\lambda \phi(x)|$ where $\lambda$ ranges over the continuous dual of $Y$.

For $X$ LF and $Y$ quasi-complete, either topology is quasi-complete.

For $X$ finite-dimensional, the stronger of these gives ${\rm Hom}(X,Y)$ a topology with features very similar to $Y$. E.g., Frechet for Frechet.

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Instead of topologicing $Hom(X,Y)$ it is common to resort back to the Gâteaux derivative and see the derivative as a map $df:U \times X \to Y$ instead of $df:U \to Hom(X,Y)$, see – Tobias Diez Sep 11 '13 at 20:12

Murat Güngör: "... 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 ..."

Considering maps $f:E\supseteq U\to F$ where $E,F$ are real locally convex spaces and $U$ is open in $E$, for about a half hundred years there have been many reasonable but generally inequivalent definitions for such a map to be smooth. However, when $E$ and $F$ are Fréchet spaces, all these "reasonable" definitions give the same concept. The Frölicher−Kriegl−Michor approach is among these "reasonable" ones. When $E$ is finite-dimensional, the "reasonable" definition of smoothness is precisely the same as in the case where $F=\mathbb R$. That is, one requires $f$ to posses continuous partial derivatives $\partial^{\kern.6mm\alpha}f$, defined in the classical manner, for all multiindices $\alpha\in\mathbb N_0^{\kern.6mm n}$ where $n$ is the dimension of $E$.

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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.

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