Let $G$ be a Banach-Lie group with Lie algebra $\mathfrak g$
and $\mathfrak h$ a closed subalgebra. Using the exponential
map and the Baker-Campbell-Hausdorff-formula one constructs a local
Lie group $\mathcal L\_H$ corresponding to $\mathfrak h$. Then the subgroup 
$H = \langle \mathcal L\_H\rangle$
of $G$ naturally inherits a topology and smooth structure (one imposes that 
$\mathcal L\_H\to H$ is an open smooth embedding)
which makes it into a Lie group and the inclusion $H \to G$ a continuous
homomorphism and a smooth immersion. In finite dimensions, this is the concept of analytic 
subgroup of Chevalley, and in infinite dimensions can be found in B. Maissen, 
Lie-Gruppen mit Banachr\"aumen als Parameterr\"aume (German),
Acta Math.\ \textbf{108} (1962) 229–270. Note that I'm avoiding on purpose use of 
the Frobenius theorem which in infinite dimensions requires that distributions 
not only be closed but also complemented. 


Since $H$ is locally arcwise connected, it is clear that its topology is
stronger that the locally arcwise connected topology 
(say denoted by $H\_\*$) constructed from the induced topology from $G$. 


(More explicitly,
a basis for the topology in $H\_\*$ is constructed by taking the arcwise connected
components of the elements of a basis for the induced topology.) 


My question is that I think both topologies coincide but
so far I can prove it only in the finite-dimensional case
(namely, I use the fact that a continuous bijective homomorphism
of a locally compact, second countable group into a locally
arcwise connected group must be a homeomorphism).
Does anyone have an argument to see that the identity map $H \to H\_\*$ is a
homeomorphism in general? Of course, it suffices to see
that $\mathcal L\_H$ is a neighborhood of the identity in $H\_\*$.



Integral subgroup theorem for Banach-Lie groups
I have been told by Karl-H. Neeb of the following neat counterexample due to K. H. Hofmann and S. Morris. Consider the Banach-Lie group $\mathfrak g=\ell^1(\mathbf R,\mathbf R)\times\mathbf R$ with the addition and take the closed discrete subgroup $D$ generated by the 
elements $(n,\sum\_{x\in\mathbf R}n(x)x)$ where $n:\mathbf R\to\mathbf Z$ is an almost null 
function. Then the quotient $G=\mathfrak g/D$ is a Banach-Lie group with Lie algebra $\mathfrak g$. Consider the subalgebra $\mathfrak h = \ell^1(\mathbf R,\mathbf R)$
and the analytic subgroup $H=\exp\_G \mathfrak h$ of $G$ it generates. Since $\mathfrak h+D=\mathfrak g$, we have that $H=G$ as a set. Hence the Lie topology on $H$ is much finer than the locally arcwise connected topology constructed from the induced topology. 


On the other hand, in the separable case there are some results showing that we can avoid 
such anomalies. 


