Resumen de On real analytic Banach manifolds

Imre Patyi, Scott Bradford Simon

• Let X be a real Banach space with an unconditional basis (for example, X = 2 Hilbert space), let �¶ �¼ X be open, let M �¼ �¶ be a closed split real analytic Banach submanifold of �¶, let E �¨ M be a real analytic Banach vector bundle, and let AE �¨ M be the sheaf of germs of real analytic sections of E �¨ M. We show that the sheaf cohomology groups Hq(M,AE) vanish for all q  1, and there is a real analytic retraction r : U �¨ M from an open set U with M �¼ U �¼ �¶ such that r(x) = x for all x �¸ M. Some applications are also given, for example, we show that any infinite-dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.