Is the image of a scaled unit ball equal to the scalar times image of unit ball?
Well of course this is true by a direct calculation: let $y\in T(B(0,R))$, so $y=Tx$ for some $x\in X$ with $\|x\|<R$. Then $\|\frac{x}{R}\|<1$ so $\frac{y}{R}=\frac{1}{R}Tx=T(\frac{x}{R})\in T(B(0,1))$ and thus $y\in R\cdot T(B(0,1))$. This shows the one inclusion. Conversely, if $y=R\cdot Tx$ for some $x\in X$ with $\|x\|<1$, then $y=T(Rx)$ and $\|Rx\|=R\|x\|<R$, so $y\in T(B(0,R))$ and this shows the other inclusion.