I was reading a proof of the following theorem.
Theorem 20.28 (Ideals and Normal Subgroups). Let $G$ be a connected Lie group,
and suppose $H \subset G$ is a connected Lie subgroup. Then $H$ is a normal subgroup of
$G$ if and only if $\operatorname{Lie}(H):=\mathfrak{h}$ is an ideal in $\operatorname{Lie}(G):=\mathfrak{g}$.
At some point it is proven that $(ad X)^kY\in \mathfrak{h}$ for all $k$. From this the person writing the proof sais $\sum\limits_{k=0}^{\infty}\frac{1}{k!}(adX)^kY\in \mathfrak{h}$. But why is this still true in the limiting process? Are we using the fact that a finite dimensional subspace of a vector space is closed? So maybe if $G$ has infinite dimension this result is not true anymore?
Best Answer
Yes, the idea is to use the fact that vector subspaces of finite-dimensional real vector spaces are always closed subsets. In infinite dimension, I suppose that it depends upon how is it that you define infinite-dimensional Lie groups.