Banach Lie algebroids and Dirac structures

We consider the category of anchored Banach vector bundles and we discuss the notion of semispray. Adding on the set of sections of an anchored Banach vector bundle a Lie bracket with some properties one gets the notion of Lie algebroid. We prove that the Lie algebroids form also a category. A Dirac structure on a Banach manifold M is defined as a subbundle of the big tangent bundle TM [direct sum] T* M that equals its orthocomplement with respect to the standard neutral metric and is closed with respect to the Courant bracket. Various characterizations of this closeness are provided. We show that with a convenient anchor any Dirac structure becomes a Banach Lie algebroid. Some examples are included. M.S.C. 2010: 53D17, 58A99. Key words: Banach Lie algebroids; Dirac structures.