Suppose that, for some finite-dimensional real vector space $\Bbb R^n$, that $n_1(v)$, $n_2(v)$, ..., $n_k(v)$ are a set of norms on the space.
Given some $v$, then, we can look at the "vector of norms", which I will denote $v_n = (n_1(v), n_2(v), ..., n_k(v))$.
We can then look at norms on this vector of norms. For example, we could take the $\ell_1$ norm of the vector, which would be the sum of norms. It is easy to see that this will also be a norm.
- Question 1: Is any norm on this "vector of norms" also a norm?
- Question 2: Likewise, if we replace with seminorms, is any seminorm on the "vector of seminorms" also a seminorm?
- Question 3: If not, for what norms do these things hold? (Do they at least hold for $\ell_p$ norms on the vector of norms?)
It is easy to see that you get homogeneity and positive-semidefiniteness, so the question is really about convexity. Does taking a "norm of norms" preserve convexity? Equivalently, does taking a norm of convex functions preserve convexity, or does taking a strictly increasing multivariate convex function of multiple convex functions preserve convexity?
EDIT - as per the answer from "mihaild" below, this isn't true for general norms, but would still like to know when it is true (in particular if it's true for $\ell_p$ norms without changing the basis).