International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 3, Pages 509-530

Differentiable semigroups are Lie groups

John P. Holmes1 and Mitch Anderson2

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 [1] in 1938. If S is a local semigroup with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at 1, then S 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.