How can we replace $\sup$ with $\max$ in definition of subordinate norm for finite-dimensional vector space?

normed-spaces

Solution 1:

First, see that you only need to consider vectors of norm $1$,

$$\lVert A\rVert = \sup_{\lVert x\rVert = 1} \lVert Ax\rVert.$$

Then note that the unit sphere in a finite-dimensional space is compact, and the function $x\mapsto \lVert Ax\rVert$ is continuous. A continuous real-valued function attains its maximum on a compact set, hence the supremum is a maximum in that case.

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