Inequality between operator norm and $\infty$-norm

Question

For a matrix $A$, is there any relation between its operator norm and $\infty$-norm (defined as the maximum of the absolute value of all the entries)?

Answer 1

The fact:

Any norms in a given finite-dimensional vector space are equivalent.

And it is easy to check the space of $n\times m$ matrices is a finite-dimensional vector space; also the operator norm and infinity norm are really norms.

So they are equivalent!
i.e.

$$||A||\le C_1||A||_{\infty}$$ and $$||A||_{\infty}\le C_2||A||$$

and that might be the relation you want.