I've just proven that for every infinite-dimensional space with a norm $(V, ||~||)$ we can find a discontinuous linear functional $ \phi $. But next I'm trying to prove the following:
The norm $ ||~|| $ is not equivalent to the norm $ |||~||| = ||~|| + |\phi| $ (meaning that identity from $ (V,||~||) $ to $ (V,|||~|||) $ is not a homeomorphism.
I can't find what might be wrong: differentiality or continuity? I was thinking that identity can't be differentiable in the point of discontinuity of $ \phi $, but I can't seem to show it correctly
If these two norms would be equivalent, there would exist a constant $c>0$ such that $$ |||x||| = \|x\| + |\phi(x)|\le c\|x\|\quad \forall x\in V. $$ This would imply boundedness of $\phi$ a contradiction. So continuity goes wrong.