Suppose we have a set $S$ which contains all functions $v \in C^{1}[0,1]$ so that $v(0) = 0$ and
$\int_{0}^{1} |v'(x)|^2 dx = 1.$
How can I show that $S$ is bounded with the infinity norm. That is, how can I show that
$ sup_{v\in S}\lVert v\rVert_{\infty} $
exists.
I know that for each $v \in S$ there exists a constant $M(v)$ so that $|v(x)| \le M(v)$ for all $x \in [0,1]$ since $v$ is continuous on a closed interval $[0,1]$. What I fail to see is how $\int_{0}^{1} |v'(x)|^2 dx = 1$ helps me.