Contents

# Contents

## Statement

###### Proposition

Let $G, H \,\in\, Grp(SmthMfd) \xrightarrow{\;Grp(undrl)\;} Grp(TopSp)$ be Lie groups (finite dimensional) with underlying topological groups denoted by the same symbol.

Then every continuous group homomorphism $G \xrightarrow{\;} H$ is smooth.

In other words, the function of hom-sets

$Grp(SmthMfd) (G,\,H) \xrightarrow{undrl_1} Grp(TopSp) (G,\,H)$

is a bijection.

This follows by applying Cartan's closed subgroup theorem to the graph of the homomorphism.

## References

Textbook accounts:

Lecture notes:

Last revised on September 3, 2021 at 07:16:31. See the history of this page for a list of all contributions to it.