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# Operator Norm, Intuitively

If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be **bounded** if $\|T\| <\infty$ where
$$\|T\|=\sup_{\underset{x\neq 0}{x\in X}}\left\{\frac{|T(x)|}{|x|}\right\}.$$
(Note that $|T(x)|$ is the norm *in $Y$* whereas $|x|$ is the norm *in $X$*.) One can show that this is equivalent to
$$\|T\|=\sup_{x\in X}\{|T(x)|:|x|=1\}.$$ So intuitively (at least in two dimensions), we can think of $\|T\|$ this way:

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