Is there a norm equivalent to the $\sup\{Lx : |x|=1\}$ norm for linear maps?

Question

One way a norm can be defined for linear maps $L(U,V)$ is taking the sup of $Lx$ where the norm of $x=1$ for $x \in U$. Another norm for linear maps is the Frobenius norm where you simply sup the square of every entry of the "matrix" that represents the linear map and take its square root. Now I am quite sure these two norms are not equivalent norms because the first norm requires an inner product defined in both the domain and codomain vector space while we don't need that for the Frobenius norm. Also, topologically speaking, the set of invertible maps is an open subset of the set of all linear maps. What I also learned recently is that the set of all positive definite maps is open as a subset of all self-adjoint maps under the first norm. I am no sure if there two subsets are considered open under the Frobenius norm, but how can we prove it? Also, are there any norms that is equivalent to the sup norm that I mentioned here?

Answer 1

If $U,V$ are finite-dimensional, then so is $L(U,V)$ and all norms on it will be equivalent, so it doesn't matter whether you use the Frobenius norm or the sup-norm. They'll have the same notion of open sets etc.