In one statement of the Weierstrass M-Test I saw the following definition:
$$\lVert f_k \rVert_\infty=\sup\{\lVert f_k(x) \rVert_2\} $$
But is this really the case?
The full statement looks like this:
Let $f_k$ be an infinite sequence of functions in $C([K\subset \mathbb{R}^n,\mathbb{R}^m])$, where $K$ is compact. If for each $k$,
$$\lVert f_k \rVert_\infty=\sup\{\lVert f_k(x) \rVert_2:x\in K\} \le M_k$$
and the series $\sum M_k$ converges, then $\sum f_k$ converges uniformly on $K$.