I want to show, that the following spaces are Banach spaces: $X_1:=\{M=(M_t)_{0\le t \le T} ;\mbox{ M is an adapted RCLL process }\}$ with the norm $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ where $P$ is a probability measure on a probability space. And $ X_2:=\{M=(M_t)_{0\le t \le T}; \mbox{M is a optional process}\}$ with the norm $\|M\|_{X_2} :=(E(\int_0^T|M_s|^2ds))^{\frac{1}{2}}$
For $X_1$ any Cauchy sequence would be a Cauchy sequence uniformly in $t$ in $L^2$. Hence there is a limit. But how do I show that this limit is again in $X_1$? Also I have no idea how to prove this for $X_2$