Show that (|M|_{mathcal{M}_{T}^{2}}:=left(mathbb{E}left[sup _{s leqslant T}left|M_{s}ight|^{2}ight]ight)^{frac{1}{2}}) is a norm in the family (mathcal{M}_{T}^{2}) of equivalence classes of

Question:

Show that $\|M\|_{\mathcal{M}_{T}^{2}}:=\left(\mathbb{E}\left[\sup _{s \leqslant T}\left|M_{s}ight|^{2}ight]ight)^{\frac{1}{2}}$ is a norm in the family $\mathcal{M}_{T}^{2}$ of equivalence classes of $L^{2}$ martingales ( $M$ and $\widetilde{M}$ are said to be equivalent if, and only if, $\sup _{0 \leqslant t \leqslant T}\left|M_{t}-\widetilde{M}_{t}ight|=0$ a.s.).

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