  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.
