# Geometric intuition for a condition on a linear operator

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?

• It might help to know that when your Banach space has an inner product, $T^\vee$ coincides with the adjoint $T^*$. – Omnomnomnom Oct 20 at 15:39
• @Omnomnomnom in the presence of an inner product the assertion becomes much more intuitive: $T$ is not surjective iff its image misses a 1d subspace, and then we can view the orthogonal projection onto this subspace as a functional for which $\|\varphi\circ T\|=0$. – Arrow Oct 20 at 16:04