Closest point projection of manifolds in Banach spaces

Suppose that $Y$ is an infinite dimensional Banach space, with an embedded finite dimensional compact submanifold $X$. It is well known (cfr Lang's Differential and Riemannian Manifolds, Thm. 5.1) that $X$ has a tubular neighborhood in $Y$, consisting in a vector bundle $\pi: NX \to X$ and a homeomorphism $f: Z \to U$, where $Z$ is a neighborhood of the zero section $\zeta X$ in $NX$ and $X \subset U \subset Y$ is an open set, satifying $f = \zeta^{-1} \circ \iota$, where $\iota: X \to Y$ is the canonical inclusion.

Under these hypothesis, is it true that a closest point projection from $U$ to $Z$ is well-defined?

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I just removed some errors in the text. –  Jaques Aug 23 '12 at 23:22
What is $NX$ and where does it reside? –  timur Aug 23 '12 at 23:41
$NX$ is the normal bundle of $X$: the Theorem is formulated in terms of the existence of a vector bundle but (at least it seems to me from the proof) one can prove that the normal bundle satisfies the thesis. –  Jaques Aug 24 '12 at 9:29