homomorphisms of Banach spaces as topological groups

Question

Let $B_1$ and $B_2$ be real Banach spaces. Verify that

i) $B_1$ and $B_2$ with the topologies by their norms are topological groups

ii) if $T:B_1 \to B_2$ is a continuous homomorphism of topological groups then $T$ is a continuous linear transformation.

Answer 1

(1) is the same that proving the continuity of sum operation.

Hint for (2): prove that for all $r=p/q\in\Bbb Q$: $$T(rx) = r T(x).$$

Particular case: Cauchy's functional equation.