# Compactness in $C_0(\mathbb R^+; X)$

Let $D\subset C_0(\mathbb R^+; X)$, where $X$ is a Banach space ($C_0$ contains continuous functions which vanish at infinity with sup norm). If $\pi_T(D)$ is compact for all $T>0$ ($\pi_T$ is the restriction operator on interval $[0,T]$), is $D$ compact?

-
Let $X=\Bbb R$. For each $n\in \Bbb N$, let $f_n$ be a continuous function such that $f_n(x)=0$ for $x\notin[n,n+1]$ and $f_n(n+1/2)=1$. Consider $D=\{ f_n :n\in \Bbb N\}$. –  David Mitra Feb 15 '13 at 18:11
Thank you very much, David. I see the negative answer. –  Kai Chen Feb 17 '13 at 8:25