What information does the operator norm provide?

Question

I am trying to improve my intuition for the operator norm (of bounded linear transformations between normed spaces). The definition $\sup_{\|x\| = 1} \|Tx\|$ is tells me that $\|T\|$ bounds the magnitude of transformed unit vectors. But I feel slightly dissatisfied because even after taking a course in bounded linear operators, my intuition is lacking.

Could you provide an alternative way to interpret the operator norm or more background such that I can appreciate its rôle in linear analysis?

Answer 1

Here's another one: the norm $M$ of the operator $T$ is the smallest number for which the assertion $\lVert Tx\rVert\leqslant M\lVert x\rVert$ holds for each vector $x$.