Equicontinuity of a set of functions: contrexemple

Question

Let $E$ be a Banach space, and $I=[0,1]\subseteq \mathbb{R}$. My goal is to construct a set of functions $X\subseteq\mathcal{C}(I,E)$ such that:

$X$ is equicontinuous but not relatively weakly compact. Any idea

Answer 1

Assume $E$ is not the zero vector space (if $E$ is the zero vector space, then so is $\mathcal{C}(I,E)$, and no such $X$ can exist).

Take $X$ to be the set of all constant functions. Then $X$ is trivially equicontinuous, since $|f(x)-f(y)|=0$ for all $f\in X$ and $x,y\in I$. But $X$ is not relatively weakly compact, since it's not bounded.