Order of infinite dimension norms I know that
$$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$
But I don't know where to put in this chain this norm:
$\|{f}\|_{L^{\infty}}=\inf\{C\in\mathbb{R},\left|f(x)\right|\leq{C}\text{ almost everywhere}\}$
Thanks pals.

If the norm $\mathcal C^1$ is defined by $\sup_{0<x<L}|f(x)|+\sup_{0<x<L}|f'(x)|$, then we have 
$$\lVert f\rVert_{L^2}\leqslant \sqrt L\lVert f\rVert_{L^{\infty}}\leqslant \sqrt L\lVert f\rVert_{\mathcal C^1}.$$
