= Solution
Write $N_2=\|M_\infty\|_2$ and $N_*=\|\sup_{t\geq0}|M_t|\|_2$. The <L2 martingale convergence theorem> ensures the existence of $M_\infty$ and yields $|M_\infty|\leq\sup_t|M_t|$ <almost surely>. Hence $N_2\leq N_*$.
For a finite horizon $T$, the <Doob L2 maximal inequality> for the <submartingale> $|M|$ gives
$$
\mathbb E\sup_{0\leq t\leq T}|M_t|^2\leq4\mathbb E|M_T|^2.
$$
The inequality does not require $M_0=0$. Send $T$ to infinity: the left side increases to $N_*^2$ by the <monotone convergence theorem>, and the right side tends to $4N_2^2$ by convergence in the <Lebesgue space> $L^2$. Thus the <equivalent terminal and maximal norms for L2-bounded continuous martingales> satisfy
$$
\boxed{\|M_\infty\|_2\leq\left\|\sup_{t\geq0}|M_t|\right\|_2\leq2\|M_\infty\|_2.}
$$
Both are genuine <norms> on <L2-bounded continuous martingales> modulo <indistinguishability of stochastic processes>: a zero terminal norm gives $M_t=\mathbb E[M_\infty\mid\mathcal F_t]=0$, first at rational times and then at all times by continuity. This proves that the two <norms> are <equivalent norms>.
Back to article page
