Integrability on Direct Limits of Banach Manifolds

Abstract

In this paper, we study several objects in the framework of direct limits of anchored Banach bundles over particular convenient manifolds (direct limits of Banach manifolds). In particular, we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anchor ranges.Proposition 14. Let (X n ) n∈N * be an ascending sequence of topological spaces. Equip X = n∈N * X n with the final topology with respect to the inclusion maps ε n : X n −→ X (i.e. the DL-topology). Then we have:O n is open in X and the DL-topology on O = lim − → O n coincides with the topology induced by X. 3. If each X n is locally compact, then X is Hausdorff. 4. If each X n is T 1 and K ⊂ X is compact, then K ⊂ X n for some n.Unfortunately, in general, a direct limit of Hausdorff topological spaces is not Hausdorff (see [Her] for an example of such a situation). Sufficient conditions on

An integrability criterion for a projective limit of Banach distributions

“…In a more general infinite dimensional context as distributions on convenient manifolds or on locally convex manifolds, in general the local flow for a vector field does not exist. Analog results exists in such previous settings: [12], [13], [8], [21] [2] for instance. But essentially, all these integrability criteria are proved under strong assumptions which, either implies the existence of a family of vector fields which are tangent and generate locally a distribution and each one of these vector fields have a local flow, or implies the existence of an implicit function theorem in such a setting.…”

Prolongations of convenient Lie algebroids

“…Remark 14. From the relation (6), it is clear that the Lie derivative of a function is defined for any section of A U . Of course, this is also true for any k-form on a Lie algebroid.…”

“…Unfortunately, in this setting, there exist many problems to generalizing all canonical Lie structures on a finite dimensional Lie algebroid and, at first, a nice definition of a Lie bracket (cf. [6]). In this paper, we consider the more general convenient setting (cf.…”

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

“…cit., GL((F n ) n∈N ) is denoted by GL(E), with E := n∈N E n . 12 A locally convex space is called a Silva space or (DFS)-space if it is a locally convex direct limit lim −→ E n for an ascending sequence E 1 ⊆ E 2 ⊆ · · · of Banach spaces, such that all inclusion maps E n → E n+1 are compact operators.…”