[Math] Example of two norms on same space, non-equivalent, with one dominating the other

Question

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other…

Answer 1

Take for instance the space $\mathcal C([0,1])$ of continuous functions on $[0,1]$. The $\mathbb L^2$ norm is dominated by the uniform norm but they are not equivalent, since convergence in $\mathbb L^2$ does not imply uniform convergence.