The space of right-continuous square-integrable martingales on a probability space $(\Omega,\mathcal A,\operatorname P)$ is a Banach space when equipped with $$\left\|M\right\|:=\sup_{t\ge0}\operatorname E\left[|M_t|\right]=\sup_{t\ge0}\left\|M_t\right\|_{L^1(\operatorname P)}.$$
Is there a commonly used name for this norm? If not, is there a terminology for norms arising in this way in general? Obviously, this space can be treated as a subspace of $L^1(\operatorname P)^{[0,\:\infty)}$. So, if $I$ is a index set and $E$ is a normed space, is there a name for the norm on $\left\{x\in E^I:\sup_{i\in I}\left\|x_i\right\|_E<\infty\right\}$ given by $\sup_{i\in I}\left\|x_i\right\|_E$?