When we have vector spaces $V,W$ over the real or complex numbers, $||\cdot||_W$ as a norm of $W$ and a linear injective mapping $f: V \to W$, we can see with little effort that $||v||^f_V=||f(v)||_W$ defines a norm on $V$.
My question is: Does this special norm have a name? And has anyone an idea when a norm $||\cdot||$ is induced in this way, i.e. there exists $g: V \to W$ linear and injective with $||\cdot||=||\cdot||^g_V$?
Thanks for your interest and thoughts in advance
Edit: $(W,||\cdot||_W)$ can not be choosen
I think I make an example:
Does there exist a linear mapping $A: \mathbb{R}^2 \to \mathbb{R}^2$ (or matrix in $\mathbb{R}^{2,2}$) with $$ ||\cdot||_1 = ||\cdot||_2 \circ A$$ or $$ ||\cdot||_2 = ||\cdot||_1 \circ A\text{ ?}$$ Do we have equivalent criteria to answer this question (for general vector spaces)?