Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that if $C(E)$, equipped with some proper metric, is a complete and seperable?
I am considering a metric on $C(E)$ which is similar to the Frechet distance: $$|F|_n \triangleq \sup_{|u| \leq n}|F(u)| \in [0, \infty]; $$ $$|F| \triangleq \sum_{n=1}^\infty 2^{-n}\frac{|F|_n}{1+|F|_n}, $$ where we set $\frac{|F|_n}{1+|F|_n}=1$ in case that $|F|_n=\infty$.
Equipped with this metric, I believe that $C(E)$ becomes a complete metric space. But traditional way of taking polynomials seems not working well for separability.
I am appreciate if you tell me the answer. If it does not work well with $C(E)$, answers to similar questions for $C_b(E)$ (bound continuous functions) or $C_L(E)$ (Lipchitz continuous functions) also help me alot.