Why supremum not max in definition of matrix norm?

Question

I know the definitions of max and sup. But, I don't see why would we use sup in this definition for example, rather than max.

This is definition of vector-induced norm of a matrix.

Answer 1

In general, we use $\sup$ when $\max$ does not exists. However, in this particular case, when $n$ is finite, the set $\{x\in K^n:\Vert x\Vert=1\}$ is a compact set. Since the continuous function $\Vert Ax\Vert$ attains its maximum on a compact set, here $\max$ exists. Hence, $\sup$ and $\max$ are the same.