Fort space on R is Eberlein compact

[yhx-12243]

In addition, this make P16 (Compact), P50 (Zero dimensional) and P187 (W-space) redundant and I've not removed any of it.

It can be discussed whether to remove or keep.

[prabau]

I have asked https://math.stackexchange.com/questions/5071093/is-the-one-point-compactification-of-an-uncountable-discrete-space-eberlein-comp for a more direct and simple proof. What do you think?

In any case, I think the current description for the Eberlein compact property (https://topology.pi-base.org/properties/P000091) is completely inadequate. We should give more explanations of what is meant, and equivalent characterizations. And mention some good references for the equivalence. It seems it's a non-trivial result of Amir & Lindenstrauss.
Do you know other good references for this topic?

@yhx-12243 @Moniker1998 You know more functional analysis than I do. What do you suggest?

[yhx-12243]

In fact the reference I provide is rather simple, though, basically using the second definition and the similar proof you provided in mathse. 😃

[prabau]

If you want to keep your explanation, I don't mind. But we can also add a second paragraph with "See also {...}" referencing the direct proof.

I was also asking about your suggestions for the best way to present the notion of Eberlein compact, i.e., how to rewrite the page for P91. Can this be done in the same PR?

[Moniker1998]

@prabau I've only seen it just now, and it seems you've got plenty of responses from others.

I agree with the definition as $\Sigma^\ast$-product of real lines is inadequate. (not sure what $\Sigma^\ast$-product is, I've only encountered $\Sigma$-products)

I haven't gone as deep into functional analysis as you might think, I've only read first 3 chapters of Banach Space Theory: The Basis for Linear and Nonlinear Analysis. Eberlein compact sets are in chapter 13. I think this book is a good reference for a lot of functional analysis though.

Def. 13.18 is the definition of Eberlein compact set in terms of compact subsets of $c_0(\Gamma)$ in its weak topology. In Corollary 13.19 they show that Eberlein compact sets are compact subsets of Banach spaces in their weak topology. Proposition 14.1 gives equivalent definition as compact subset of $C_p(K)$ for some compact $K$. Also see 14.9, 14.11 for more equivalent definitions.

Perhaps some results from chapter 14 could also be added to pi-base.

They don't mention that pointwise convergence and weak convergence on $c_0(\Gamma)$ is equivalent, from what I've seen.

[prabau]

$\Sigma^\ast$-product of real lines is explained for example on p. 146 of Encyclopedia of General Topology (Hart, Nagata, Vaughan, eds.):

"Recall that the $\Sigma^\ast$-product of real lines is the set of all points $x$ of $\mathbb R^\tau$ (for some $\tau$) such that, for every $\epsilon>0$, the number of coordinates of $x$ not in the interval $(-\epsilon,\epsilon)$ is finite."

In other words, it's the same as $c_0(\Gamma)$.

[prabau]

@yhx-12243 I have added the mathse reference. Please check if it's ok. I suggest to go with this for now.

@Moniker1998, the Banach Space Theory book you mentioned seems a pretty good resource for this. When you have time, could you do a separate PR with a proposal for rewriting the P91 definition page?
We can discuss further based on that.

[Moniker1998]

@prabau ah okay so its just another variation on what coordinates are allowed. This is odd to me that they would name them $\Sigma^\ast$-products and not $\sigma^\ast$-products, since $\Sigma$ refers to all but countable amount, and $\sigma$ to all but finite amount, to my knowledge.

[Moniker1998]

@prabau I think you're usually much better than me when it comes to rewriting things and deciding what to keep and what not. I'll leave that task to you

[prabau]

That's fine. I can write up something after doing some more reading. Unless @yhx-12243 would like to do it?

[prabau]

@prabau ah okay so its just another variation on what coordinates are allowed. This is odd to me that they would name them Σ ∗ -products and not σ ∗ -products, since Σ refers to all but countable amount, and σ to all but finite amount, to my knowledge.

Yes, $\Sigma^\ast$-product is like $\Sigma$-product with an extra decaying condition. So with kind of obvious notation:
$\sigma\mathbb R^\kappa\subseteq\Sigma^\ast\mathbb R^\kappa\subseteq\Sigma\mathbb R^\kappa\subseteq\mathbb R^\kappa$,
all with topology of pointwise convergence, i.e., induced from product topology on $\mathbb R^\kappa$.