Compactness of $C[J,X]$?

Question

If $J=[0,T]\subset \mathbb{R}$, and $X$ is a Banach space? Then, is $C[J,X]$ (Banach space of all continuous functions from $J$ to $X$) a compact metric space with respect to supnorm defined as $\|x\|_{\infty}=\sup \{x(t): t\in[0,T]\}$?

Answer 1

The only normed space that is compact is $\{0\}$. Any other Banach space is unbounded (as soon as $x\ne0$, $\|nx\|$ can be made arbitrarily large for $n\in\mathbb N$), so not compact.

Answer 2

No unbounded metric space can be compact. In particular $C(X,Y)$ is not compact when $Y$ is not bounded. Just pick a sequence of constant functions, with unbounded values. And every normed space is unbounded, except for the zero space.

But even if both $X$ and $Y$ are compact, $C(X,Y)$ does not have to be. Consider $X=Y=[0,1]$ and the sequence $f_n(t)=t^n$.