International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 509-530
Differentiable semigroups are Lie groups
1Department of Mathematics (FAT), Auburn University, 36849-5310, Auburn, USA
2Department of Mathematics, University of Hawaii at Hilo, Hilo 96720-4091, HI, USA
Received 18 May 1992; Revised 25 April 1994
Copyright © 1995 John P. Holmes and Mitch Anderson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present here a modern, detailed proof to the following theorem which was introduced
by Garrett Birkhoff  in 1938. If is a local semigroup with neighborhood of homeomorphic to a
Banach space and with multiplication strongly differentiable at , then is a local Lie Group. Although
this theorem is more than 50 years old and remains the strongest result relating to Hilbert's fifth problem
in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it
is the goal of the authors here to clarify its importance and to demonstrate a proofwhich is more accessible
to contemporary readers than the one offered by Birkhoff.