I am reading Brian Hall's book 'Lie groups, Lie algberas, and Representations' and on p52, corollary 2.34 reads : " Every continuous homomorphism between two matrix Lie groups is smooth."

I am wondering if this is true for general Lie groups. If not, please provide a counterexample.

What are all the properties that hold good for matrix Lie groups (and their representations) but not for general Lie groups ? It would be very helpful for me to have a list.

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    $\begingroup$ The result is true in general: every continuous homomorphism between Lie groups is automatically smooth. With regards to your general question, you might find this mathoverflow thread useful: mathoverflow.net/questions/81610/… $\endgroup$ – Eric O. Korman Nov 8 '13 at 4:23

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