Let $T:V\to W$ be a linear operator between Banach spaces. Write $$\nu(T)=\inf_{\|\varphi\|=1}\| T^\vee(\varphi) \|$$ where $\varphi\in W^\vee$ and $T^\vee$ is precomposition with $T$.
Question 1. What is the geometric meaning of $\nu(T)$?
I've also read that the uniform boundedness principle implies $\nu(T)>0\iff T$ is surjective.
Question 2. How to prove this?
Question 3. If $W$ is a Hilbert space, can $\nu(T)$ be calculated by taking the infimum over orthogonal projections onto lines?