Let $\mathbb S^2(0,T)$ be the set of pregressively measurable processes s.t : $$ Y\in \mathbb S^2(0,T)\ \Leftrightarrow\ \parallel Y\parallel_{\mathbb S^2(0,T)}=\mathbb E (\sup_{t\in[0,T]} \mid Y_t\mid^2)^{1/2}<+\infty $$
Let $\mathbb H^2(0,T)$ be the set of pregressively measurable processes s.t : $$ Z\in \mathbb H^2(0,T)\ \Leftrightarrow\ \parallel Z\parallel_{\mathbb H^2(0,T)}=\mathbb E (\int^T_0 \mid Z_t\mid^2dt)^{1/2}<+\infty $$
Do we have $\mathbb S^2(0,T)=\mathbb H^2(0,T)$ ?
Does this following argument enough for stipulating the equality ? $$ \sup_{t\in[0,T]} \mid X_t\mid^2 \leq \int^T_0 \mid X_t\mid^2dt\leq T\sup_{t\in[0,T]} \mid X_t\mid^2 $$