I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric $$d(f,g)=\sup_x d(f(x),g(x)).$$ Can we say the space $C(X,X)$ is compact? In the special case which I need my space $X$ is also complete and separable.