# What is the definition of being smooth for a function from a Lie group to a Fréchet space?

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 $V$?

• There is a book by A.Kriegl and P.Michor, "The Convenient Setting of Global Analysis" abebooks.com/9780821807804/… 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.
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 $E$.