Projective limits of Banach-Lie groups (Q1925021)

It is known that for Fréchet-Lie groups there is no exponential map in general as in the case of Banach-Lie groups. One can construct, however, exponential maps on certain classes of Fréchet-Lie groups, for example Omori strong ILB-groups. The author studies the inverse situation for certain Fréchet-Lie groups. He proves that if a commutative Fréchet-Lie group \(G\) has an exponential map which is a local diffeomorphism at \(e\in G\), then \(G\) is the limit of a projective system of Banach-Lie groups.