  Let $X, Y$ be separable metrizable spaces, where $X$ is noncompact and $Y$ is
equipped with an admissible complete metric $d$. We show that the space
$C(X,Y)$ of continuous maps from $X$ into $Y$ equipped with the uniform
topology is locally homeomorphic to the Hilbert space of weight $2^{\aleph_0}$
if (1) $(Y, d)$ is an ANRU, a uniform version of ANR and (2) the diameters of
components of $Y$ is bounded away from zero. The same conclusion holds for the
subspace $C_B(X,Y)$ of bounded maps if $Y$ is a connected complete Riemannian
manifold.
