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I have been given the definition of a subordinate (operator or matrix) norm: $$\lvert\lvert A \rvert\rvert=\sup_{x\in V}\frac{\lvert\lvert Ax \rvert\rvert}{\lvert\lvert x \rvert\rvert},$$ where $V$ is a vector space, and $A$ is a linear operator. I have been told that in the case of a finite-dimensional vector space, we can replace $\sup$ with $\max$, giving $$\lvert\lvert A \rvert\rvert=\max_{x\in V}\frac{\lvert\lvert Ax \rvert\rvert}{\lvert\lvert x \rvert\rvert},$$ and I am wondering why this is true. It seems reasonable to me, but how can I be convinced that it's true?

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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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