Let $X$ and $Y$ be two Banach spaces under the same field $K = \mathbb{R}$ or $K = \mathbb{C}$, and $T: X \rightarrow Y$ is a linear operator. Show that
(a) if $B(y, \epsilon) \subset T(B(x,\delta))$, then $B(\alpha y, \alpha \epsilon) \subset T(B(\alpha x,\alpha \delta))$, where $\alpha >0$.
(b) if $B(y, \epsilon) \subset \overline{T(B(x,\delta))}$, then $B(\alpha y, \alpha \epsilon) \subset \overline{T(B(\alpha x,\alpha \delta))}$, where $\alpha >0$.
I have doubts in both cases. I read the lemma in Kreyszig's book to understand the Open Mapping Theorem, but I couldn't solve this problem and I don't understand why $X$ and $Y$ need to be Banach spaces, that is, the result continues valid for $X$ and $Y$ normed spaces?