Is the problem of calculating the induced norm of a linear operator (in a finite or infinite-dimensional space) generally a difficult one ?
And by difficult I mean, that there are no closed formulas or no general procedure that always yields the induced norm.
Of course, for the usual spaces with the usual norms, there are formulas, that makes ones life very, so one can take shortcuts in calculating the induced norm of a operator (instead of trying to use the definition of the induced norm : $ ||A||=\sup_{||x||=1} ||Ax||).$ But is there also a procedure how to calculate $||A||$ for some very weird vector norms, or for some unsual infinitedimensional spaces (since in finite dimensions we can at least use the fact, that every vectors space is isomorphic to $\mathbb{K}^n$ for some $n$) ?
EDIT: I think the user tomasz bst described what I meant. Are there vector norms such that for their induced operator norm it is proven that there isn't a closed expression ?