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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?

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    $\begingroup$ Since $T$ is linear, it isn't useful to let $$||T||:= \inf_{x} |T(x)|,$$ because given any $x \neq 0$ you can scale by $c$ so that $|T(cx)|$ is a large as you like. (If that were the definition, then $||T||= \infty$.) Thus, the norm of $T$, however it is defined, should somehow be dependent on $x$. $\endgroup$ – Dzoooks Jul 4 '19 at 22:48
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    $\begingroup$ There isn't really any deeper intuition. Just the 'worst case' scaling. $\endgroup$ – copper.hat Jul 5 '19 at 2:46
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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 $x$.

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    $\begingroup$ And this $M$ measure the 'contractness' of the operator. $\endgroup$ – Gonzalo Benavides Jul 4 '19 at 22:33

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