Space of Adapted Continuous Process is a Banach space

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.

• No, I don't think so. If $(X_n)_{n \in \mathbb{N}} \subseteq \mathbb{S}^2$, then $X=\lim_n X_n$ exists in $L^2$-sense but there is no reason why $X$ should have continuous sample paths with probability 1. – saz Nov 4 '18 at 12:34
• Thank you. I forgot one detail in the definition of $\mathbb{S}^2.$ But, I guess this does not make any difference. – White Nov 4 '18 at 12:39
• No, it doesn't make a difference. – saz Nov 4 '18 at 12:51

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 $$t . Since $$Y_t$$ does not have continuous paths the space is not complete.