Function spaces of CW homotopy type are Hilbert manifolds

Let $X$ be a countable CW complex and $Y$ an ANR (for metric spaces) and let $Y^X$ denote the space of continuous maps from $X$ to $Y$ with the compact-open topology. We show that, under mild restrictions, the following are equivalent: (1) $Y^X$ is an $\ell^2$-manifold, (2) $Y^X$ is an ANR, (3) $Y^X$ has the homotopy type of a CW complex. We also give a few interesting examples and applications.