I am studying functional analysis.I have observed that if I do not get an intuitive idea of what a theorem/result is saying,then we have to remember it by heart.Now something similar is happening to the theorem about norm of a self-adjoint operator.We have a result that $\|T\|=\sup\limits_{\|x\|=1}|\langle Tx,x\rangle|$ and I want to visualize what it means.I like to think in the following way:
We look at all $x$ on the unit circle/sphere $S=\{x\in H:\|x\|=1\}$ and we look at the image of the vector $x$ under the map $T$.If $T$ is self-adjoint then $\|T\|$ is basically the maximum value of the projection of $Tx$ along $x$.
But I want to understand why this is true.Can someone provide me some motivation?