I Why Does Weak-* Topology Use Finite Neighborhoods?

[Samy_A]

Aargh, sorry! I just realized I made a typo in my post #28. The proposition should have said:

Proposition:
For every weak-* continuous antilinear functional $\Psi$ on $H^\times$, there is a vector $\psi\in H$ such that $$\Psi(\phi) ~=~ \phi(\psi) ~,~~~~~ \mbox{for all}~ \phi\in H^\times ~.$$I.e., the difference is that $\phi$ can be any element of $H^\times$, not merely $H$.

There must be a typo here, since $\Psi(\phi)$ is antilinear in $\phi$ and $\phi(\psi)$ is linear in $\phi$.

It could be as follows:
Proposition:
For every weak-* continuous antilinear functional $\Psi$ on $H^\times$, there is a vector $\psi\in H$ such that $$\Psi(\phi) ~=~ \psi(\phi) ~,~~~~~ \mbox{for all}~ \phi\in H^\times ~.$$

I guess that means your proof must change significantly?

Well, not by much.
I already established (assuming the proof is correct) that there is a vector $\psi\in H$ such that $\Psi(\phi) ~=~ \psi(\phi) ~~~~ \mbox{for all}~ \phi\in H~.$
But $H$ is dense in $H^\times$ with the weak-* topology, so since the weak-* continuous functionals $\Psi, \psi$ are equal on a dense subset of $H^\times$, they must be equal on all of $H^\times$

 

[strangerep]

There must be a typo here, [...]

No, this time it was an actual mistake.
The original paper used $\langle\phi, \psi\rangle$, but I was trying to keep parenthesis notation.

I need more time to study your proof properly. The existing proof in the paper (which is apparently an instance of a general result: Thm 3.10 in Rudin FA) is hard to relate to yours. Later...

(Oh, and thanks yet again. )

 

[Samy_A]

No, this time it was an actual mistake.
The original paper used $\langle\phi, \psi\rangle$, but I was trying to keep parenthesis notation

I need more time to study your proof properly. The existing proof in the paper (which is apparently an instance of a general result: Thm 3.10 in Rudin FA) is hard to relate to yours. Later...

(Oh, and thanks yet again. )

Ah, Rudin's Functional Analysis (sweet memories ... )

Yes, that is definitely a faster and more elegant way to prove it. Maybe one can "map" the steps in my proof to the more general steps in Rudin's proof of theorem 3.10, but why bother.

EDIT: so I bothered anyway. After reading Rudin's proof, one can indeed see the similarities.
Where my proof diverges is that after finding the set $B \subset U$, I prove that $\Psi$ is a linear combination of $\phi_1,...,\phi_n$ by using a Pre-Hilbert space argument. Rudin uses his more general lemma 3.9 to prove that.

 

[strangerep]

I figured I should append a closing post to this thread.

I've realized that the author of the paper I originally referred to is not an FA expert. Some of the proofs therein are taken verbatim from experts without additional elaboration. Others are beyond my ability to assess without spending a vast amount of time, and even then doubts would remain.

So I think I must put the paper aside for now, and get on with other things.

Thank you again to Samy_A and Krylov. I can say that, with your help, my understanding of weak topology is now better than it was, though I still have a long way to go.