The definition is
$$\|A\| = \sup_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \max \left\{ \|Ax\|: \|x\|=1 \right\}$$
but how is maximum coming in place?
2026-03-25 09:37:02.1774431422
The operator norm is defined based on the supremum or equivalently the maximum.
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I assume here you mean that you are working in a finite dimensional normed space. In that case the sphere $\{x: ||x||=1\}$ is a compact set, and since linear transformations in finite dimensional normed spaces are continuous, this implies that $||Ax||$ has a maximum on that sphere.
This is no longer true in infinite dimensional normed spaces. There you really must write supremum, not maximum.