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?

  • $\begingroup$ It might help to know that when your Banach space has an inner product, $T^\vee$ coincides with the adjoint $T^*$. $\endgroup$ – Omnomnomnom Oct 20 at 15:39
  • $\begingroup$ @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 $\|\varphi\circ T\|=0$. $\endgroup$ – Arrow Oct 20 at 16:04

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