Let $(\Omega, \mathcal F, P, \mathcal F_t)$ be a filtered probability space. Consider two spaces $M$ and $S$ defines as follows:
$M$ is a collection of all continuous $\mathcal F_t$-adapted processes of the form $\phi: [0,1]\times \Omega \mapsto \mathbb R$ satisfying $$\mathbb E [\int_0^1 \phi_s^2 ds]<\infty.$$
$S$ is a collection of all continuous $\mathcal F_t$-adapted processes of the form $\phi: [0,1]\times \Omega \mapsto \mathbb R$ satisfying $$\mathbb E [\sup_{0\le t \le 1} \phi_t^2]<\infty.$$
[Q.] It is obvious that $S\subset M$. Can we say $S = M$? If not, find a $\phi\in M\setminus S$.