On real analytic Banach manifolds
  Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $\Omega\subset X$ open, $M\subset\Omega$ a closed split real analytic Banach submanifold of $\Omega$, $E\to M$ a real analytic Banach vector bundle, and ${\Cal A}^E\to M$ the sheaf of germs of real analytic sections of $E\to M$. We show that the sheaf cohomology groups $H^q(M,{\Cal A}^E)$ vanish for all $q\ge1$, and there is a real analytic retraction $r:U\to M$ from an open set $U$ with $M\subset U\subset\Omega$ such that $r(x)=x$ for all $x\in M$. Some applications are also given, e.g., we show that any infinite dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable. 
Imre Patyi, Scott Simon