Function spaces are sometimes not metrizable. Let $X$ and $Y$ be topological spaces, and $C(X,Y)$ be the space of continuous maps from $X$ to $Y$, topologized in the compact open topology. Then $C(X,Y)$ need not be metrizable (it is if $X$ is compact, and $Y$ is a metric space though).

Function spaces are sometimes not metrizable. Let $X$ and $Y$ be topological spaces, and $C(X,Y)$ be the space of continuous maps from $X$ to $Y$, topologized in the compact open topology. Then $C(X,Y)$ need not be metrizable (it is if $X$ is a compact, and $Y$ is a metric space, it is though).