Does there exist a vector space $X$ and two norms $\Vert \cdot \Vert$ and $\Vert \cdot \Vert_1$ on $X$ such that both spaces $(X, \Vert \cdot \Vert)$ and $(X, \Vert \cdot \Vert_1)$ are complete, but the two norms $\Vert \cdot \Vert$ and $\Vert \cdot \Vert_1$ are not equivalent?
A fact: When one of these two norms is stronger than the other one, then they are equivalent. So I want to find a counter example, where the above condition does not hold.
And I am now understand Daniel Fischer's answer.
– Lei Zhang May 03 '17 at 23:43