Lie group, Banach

A set $ G $ endowed with a group structure and an analytic Banach manifold structure (cf. Banach analytic space) at the same time; these two structures are compatible in the following sense: the mapping $ ( g , h ) \rightarrow g h ^ {-} 1 $ from $ G \times G $ into $ G $ is analytic. If the Banach manifold is finite dimensional, this concept coincides with the usual concept of a Lie group.

Examples. A Banach space with an addition operation, the set $ A ^ {*} $ of invertible elements of a Banach algebra $ A $ with a multiplication operation, and the set $ C ^ {k} ( M , G ) $ of functions smooth of order $ k $ on a smooth manifold $ M $ with values in a Lie group $ G $ and with the operation of pointwise multiplication, are Banach Lie groups. On the other hand, the set $ \mathop{\rm Diff} ^ {k} M $ of one-to-one mappings smooth of order $ k $ of a smooth manifold $ M $ onto itself is not a Banach Lie group: in this case the natural structures of the Banach manifold and the group (with respect to the operation of composition) are not compatible.

Some fundamental theorems in the theory of Lie groups remain true for Banach Lie groups: to every Banach Lie group corresponds a Banach Lie algebra, from which in turn one can recover a local Banach Lie group. It is known, however, that not every local Banach Lie group can be extended to a global one [2]; a neighbourhood of the identity of a Banach Lie group is covered by the image of the exponential mapping; there is a correspondence between connected closed subgroups of a Banach Lie group and closed subalgebras of the corresponding Lie algebra.

There are various generalizations of the concept of a Banach Lie group (see [3]), in which the Banach space structure is replaced by a linear topological space structure of more general type.

References

[1]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[2]	P. de la Harpe, "Classical Banach–Lie algebras and Banach–Lie groups of operators in Hilbert space" , Springer (1972)
[3]	H. Omori, "Infinite dimensional Lie transformation groups" , Springer (1974)