What's the difference between the operator norm and the sup norm

Question

What's the difference between the operator norm and the sup norm over $C[0,1]$. a.k.a $\left\lVert x\right\rVert_\infty$ vs $\left\lVert x\right\rVert_{op}$

Answer 1

These are two different norms for entirely different purposes.

The supremum norm over $C[0,1]$ is the norm of this particular Banach space. We have that $\|\cdot\|_{\infty}:C[0,1]\to[0,\infty)$ and this norm measures the size of a continuous function.

The operator norm is used for bounded linear operators that map the elements of one normed space to another normed space. For example, if $L$ is a bounded linear operator from $X$ to $Y$, where $X$ and $Y$ are two normed spaces, then the operator norm is defined as the smallest $M$ such that $\|Lv\|_Y\le M\|v\|_X$ (see here for more details). The operator norm measures the size of an operator.

I hope this is helpful.