If $\gamma\colon [a,b] \rightarrow \mathbb{R}^d$ is a differentiable curve we usually define $$ \text{length}(\gamma) := \int_a^b \|\dot{\gamma}(t)\|_2\, \text{d}t,$$ where $\|\cdot\|_2$ is the Euclidean norm on $\mathbb{R}^d$. My question is:
How do we define $\text{length}(\gamma)$ if we use the sup norm $\|\cdot\|_\infty$?
This is incorrect. The sup-norm is the maximum over the different coordinate components of $\gamma$ for each $t$, so you would end up with a function of $t$, $M(t)$. This would still need to be integrated over your range.