When are two norms equivalent on a Banach space?
Solution 1:
Hint: Define $\Vert \dot\, \Vert_3 := \Vert \dot\, \Vert_1 + \Vert \dot\, \Vert_2$. Show that $\Vert \dot\, \Vert_3$ is a complete norm on $E$. Now use the fact, that the map $f: (E, \Vert \dot\, \Vert_3) \to (E,\Vert \dot\, \Vert_1)$ with $f(x) = x$ for all $x\in E$ is a continuous bijection between Banach spaces.