completeness of the space $C(E,F)$

Question

Let $E$ be a compact space and $F$ a Banach space.

I want to show that $C(E,F)$ (The space of continous functions) is a complete space.

Thank you .

Answer 1

See, for instance, more general Theorem 2.3.4 here or even more general Proposition 1.2 here, stating that if $X$ is a topological space and $Y$ is a metric space then a metric space $C_b(X,Y)$ of all continuous bounded functions from $X$ to $Y$ endowed with the uniform metric is complete.