Theorem: $||.||$ and $||.||'$ norms in the vector space $E$ are equivalent $\iff \exists A,B>0 $ st. $A||x|| \leq ||x||' \leq B||x||$.
Proof: $I:(E,||.||) \to (E,||.||')$ is linear. Then $I:(E,||.||) \to (E,||.||')$ is continuous $\iff \exists B>0 $ st. $||x||' \leq B||x|| \forall x \in E$.
My problem is seeing why $I:(E,||.||) \to (E,||.||')$ is linear. How can I verify this? Linearity is defined by these equalities: $I(x+y) = I(x) + I(y)$ and $I(ax) = aI(x)$. I don't quite see where the norm enters here...
The rest of the theorem I understood because of a previous proposition regarding linear transformations, continuity, and existence of $B$.