I'm looking for some light on this problem
Let $A$ and $B$ be subspaces of a vector space $V$ such that $V=A\oplus B$, and let $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ be two norms on $V$. If $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent on $A$ and equivalent on $B$, must they be equivalent?
I know (quite sure) that the answer is negative and would like to find counter examples. Such space is obviously infinite dimensional otherwise every norm is equivalent. I think it might be wise to try to break a known space (like $l_\infty$) into a clever product that violates the bound for equivalent norms. Any help would be appreciated.
Let $X$ be an infinite dimensional nirmed space. There exists an unbounded linear functional $\varphi$ on $X.$ Fix $v\in X$ so that $\varphi(v)=1.$ Let $A=\ker\varphi$ and $B=\mathbb{C}v.$ Then $X=A\oplus B.$ Indeed, for $x\in H$ we have $$x=[x-\varphi(x)v]+ \varphi(x)v$$ Let $\|\cdot \|_1$ denote the original norm while $$\|x\|_2=\|x\|_1+|\varphi(x)|$$ The norms coincide on $A.$ They are equivalent on one dimensional subspace $B.$ However the norms are inequivalent as the functional $\varphi$ is unbounded.