(This is from the proof of the open mapping theorem, but this context isn't terribly important).
$B(X)$ means unit ball in $X$.
We have a bounded, linear, and surjective $T:X\to Y$ between Banach spaces. I have proved that $\overline{T(B(X))}$ contains ball $B_r(y)$. Now I want to exhibit a ball (with any radius) about the origin in $\overline{T(B(X))}$. I think this works:
- Set addition argument: note that $$B_{r}(0)\subseteq B_{r}(y)+\{-y\}\subseteq \overline{T(B(X))}+\overline{T(B(X))}\subseteq \overline{T(2B(X))}$$
This halves the radius and gives me $B_{r/2}(0)\subseteq\overline{T(B(X))}$ - fine. Can I avoid losing this constant by arguing from convexity? E.g.
- $B_r(y)\subseteq \overline{T(B(X))}$, so, as $T$ is linear and $B(X)$ symmetric about the origin, we have $B_r(-y)\subseteq\overline{T(B(X))}$. In fact, $\overline{T(B(X))}$ is convex, so we get $B_r(0)\subseteq \overline{T(B(X))}$.
Are both approaches valid?