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There is some (probably stupid) thing that I did not get in Serge Lang's $SL_2(\mathbf{R})$: On page 93 he considers a representation $\pi:G\to GL(H)$ of a group $G$ in a Banach space. Then he defines the subspace $H_\pi^\infty$ as $$ H_\pi^\infty = \{v\in H\ |\ x\mapsto \pi(x)v\ \text{ is }\ C^\infty \} $$ and uses this to define the derived representation.

My question simply is: Since $G$ is just a group: How is differentiability of the maps $F_v: G \to H$, $F_v(x) = \pi(x)v$ defined?

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  • $\begingroup$ The group $G$ must be a Lie group for this to make sense, so it has a differentiable structure. $\endgroup$ Jul 12, 2011 at 13:19
  • $\begingroup$ Sorry for the stupid question and thanks for the totally non-stupid answer! $\endgroup$
    – Dirk
    Jul 13, 2011 at 7:29

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