Space of Adapted Continuous Process is a Banach space

Question

We assume a complete filtered probability space with right continuous filtration.

Let $\mathbb{S}^2$ be the space of adapted and almost surely continuous processes $X = (X_t)_{t \in [0,T]}$ with values in $\mathbb{R}$ and $$ E[ \sup_{t \in [0,T] } | X_t | ^2 ] < \infty. $$ Define the norm $$ || X ||^2 = E \left[ \int_0^T |X_s|^2 ds \right]. $$

Is $\mathbb{S}^2$ a Banach space?

I know it will a Banach space w.r.t the uniform norm.

Answer 1

Let $X_t(\omega) =(\frac t T)^{n}$. This sequence converges in the norm to $Y_t(\omega) =1$ for $t=T$ and $0%$ for $t <T$. Since $Y_t$ does not have continuous paths the space is not complete.