I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces.

Rainwater's theorem. Let $X$ be a Banach space, let $\{x_n\}$ be a bounded sequence in $X$ and $x \in X$. If $f(x_n)\to f(x)$ for every $f\in\operatorname{Ext}(B_{X^*})$, then $x_n \overset{w}\to x$.

The symbol $x_n \overset{w}\to x$ denotes the convergence in weak topology. By $B_{X^*}$ we denote the unit ball of the dual $X^*$ (with respect to the usual operator norm) and $\operatorname{Ext}(B_{X^*})$ is the set of all extreme points of this set.

See e.g. Corollary 3.137, p.140 in Banach Space Theory: The Basis for Linear and Nonlinear Analysis by Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler.

In particular, if we apply the above to the space $C(K)$, where $K$ is compact, we get the following result (Corollary 3.138) in the same book.

Corollary. Let $K$ be a compact topological space. Let $\{f_n\}$ be a bounded sequence in $C(K)$ and $f\in C(K)$. Then, if $f_n\to f$ pointwise, we have $f_n \overset{w}\to f$.

Weak convergence of a sequence in $X$ means, by definition, that $f(x_n)\to f(x)$ for each $f\in X^*$. Rainwater's theorem essentially says that there is a smaller set of functionals we need to check - namely the set $\operatorname{Ext}(B_{X^*})$. It is quite natural to ask whether the same is true for nets.

  • Let $(x_\sigma)_{\sigma\in\Sigma}$ be a net in a Banach space $X$ and let $x\in X$. Is it true that $x_\sigma \overset{w}\to x$ if and only if $f(x_\sigma)\to f(x)$ for every $f\in\operatorname{Ext}(B_{X^*})$?

The weak topology is precisely the initial topology on $X$ w.r.t. all linear continuous functionals. Again, it is natural to ask whether we can replace $X^*$ by a smaller set. In this way we get a reformulation of the above question.

  • Is the weak topology on $X$ the initial topology w.r.t. $\operatorname{Ext}(B_{X^*})$?

If the answer to the above questions is negative, I would like to know whether they hold at least for $C(K)$.

  • If $(f_\sigma)_{\sigma\in\Sigma}$ is a net in $C(K)$ and let $f\in C(K)$. Is it true that $f_\sigma$ converges to $f$ weakly if and only if it converges pointwise?
  • Is the weak topology on $C(K)$ the same as the initial topology w.r.t. the maps $f\mapsto f(x)$ for $x\in K$ (i.e. the evaluations at all points of $K$)?
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    $\begingroup$ Two quick remarks: 1) A nice exposition of Rainwater's theorem is given in Phelps's book Lectures on Choquet's theorem, Chapter 5 that might also address a few of your other questions. 2) A counterexample to your penultimate question is given at the beginning to my answer here (I give an example of a pointwise convergent net of continuous functions on $[0,1]$ that doesn't converge weakly). $\endgroup$ – t.b. Aug 19 '12 at 7:19
  • $\begingroup$ @t.b. The fact that the sequence from your example does not converge weekly can be seen by using the functional $f\mapsto \int_0^1 f \,d\mu$, if I understand it correctly. That's what you meant, right? $\endgroup$ – Martin Sleziak Aug 19 '12 at 7:25
  • $\begingroup$ Yes, precisely. (I'm not sure if this counterexample answers all your questions but it looks like it). Let me know if you want me to expand on it. $\endgroup$ – t.b. Aug 19 '12 at 7:26
  • $\begingroup$ @t.b. You were correct about the missing quantifier. Of course, your counterexample special case is a counterexample for the more general claim, too. Could you please post this as an answer? (The problem does not seem that difficult now that you told me the answer.) $\endgroup$ – Martin Sleziak Aug 19 '12 at 7:31
  • $\begingroup$ Sure, I'll do that. I'll take a little time to think about whether I have anything useful to add. $\endgroup$ – t.b. Aug 19 '12 at 7:32

Unfortunately, none of your questions seems to have a positive answer, even in the most favorable circumstances, as is shown by the following example, given in my answer to A net version of dominated convergence?. It is a standard example showing that there's no hope to prove a dominated convergence theorem for nets instead of sequences (I think I learned it from lectures by O.E. Lanford, but I'm not sure).

Take $K = [0,1]$ and $X = C(K)$.

Let $\Lambda$ be the set of finite subsets of $[0,1]$, ordered by inclusion, so as to make it a directed set. For every $\lambda \in \Lambda$, choose a function $f_\lambda \colon [0,1] \to [0,1]$ such that $f_\lambda(x) = 1$ for all $x \in \lambda$ and $\int_{0}^1 f_\lambda(t)\,dt \leq \frac{1}{2}$. Observe that the net $(f_{\lambda})_{\lambda \in \Lambda}$ converges pointwise to the constant function $1$. However, $f_\lambda$ does not converge weakly to $1$ since by construction $\int (1 - f_\lambda) \geq \frac{1}{2}$ for all $\lambda \in \Lambda$.

The original reference for Rainwater's theorem is:

John Rainwater, Weak convergence of bounded sequences, Proc. Amer. Math. Soc. 14 (1963), page 999.

whose proof is the same as the one given in chapter 5 of Robert R. Phelps's Lectures on Choquet's theorem, Springer Lecture Notes in Mathematics 1757. These proofs are based on the Bishop-de Leeuw extension of Choquet's theorem given as Theorem 5.6 in

Errett Bishop, Karel de Leeuw, The representations of linear functionals by measures on sets of extreme points. Annales de l'institut Fourier, 9 (1959), p. 305–331.

A rather different proof of Rainwater's theorem (or rather a generalization of it) was given by S. Simons in

S. Simons, A convergence theorem with boundary, Pacific J. Math. Volume 40, Number 3 (1972), 703–708.

The Rainwater theorem and some closely related results (all essentially sequential in nature) is also discussed in Diestel's Sequences and series in Banach spaces, Springer GTM 92 (1984), which also gives many further pointers to the literature.

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    $\begingroup$ For background on the pseudonym John Rainwater, see the Wikipedia article; Rainwater is also mentioned on the MO thread on pseudonyms of famous mathematicians (thanks to Martin for those pointers!). $\endgroup$ – t.b. Aug 19 '12 at 8:46
  • $\begingroup$ t.b. I have a follow-up question, but don't spend too much time if you don't see an answer immediately. (I'll try to get back to this later and if I don't come up with some solution I'll post a new question.) The question is: Would the same trick work for every compact space? If we have a compact space $K$ and take the same net as in your example, is there a functional $\varphi\in C(K)^*$ such that $\varphi(f_\lambda)$ does not converge to $\varphi(f)$? If not, is it possible at least for locally compact topological groups? $\endgroup$ – Martin Sleziak Aug 19 '12 at 8:47
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    $\begingroup$ And I also wanted to thank you for your answer, which was really helpful. (And also for the wealth of references, although I am not sure how soon I'll get to reading some of them.) I'm glad you're one question closer to the specialist badge for examples-counterexamples now. (IIRC to get the bronze badge it is necessary to answer at least 20 questions.) $\endgroup$ – Martin Sleziak Aug 19 '12 at 8:52
  • $\begingroup$ I suppose what's needed is along the lines of the following: Let $X$ be a locally compact space and let $\mu$ be a Borel measure on $X$ having the following properties: there is a compact subset $K$ of pos. measure and an open rel. cpt. neighborhood $U$ of $K$ such that every point $x \in U$ has a neighborhood of arbitrarily small positive measure. Take a function $f\colon X \to [0,1]$ which is constant equal to one on $K$ and supported in $U$. Then modifying the example yields a ptw. conv. net in $C_0(X)$ that doesn't converge weakly. This should work on locally compact non-discrete groups. $\endgroup$ – t.b. Aug 19 '12 at 8:59
  • $\begingroup$ Maybe I should have added the following: the pointwise topology on $C(X)$ was investigated in very deep work by A. Bellow (= A. Ionescu Tulcea), Fremlin, Talagrand and others. It quickly leads to set-theoretic problems. See chapters 438, 439, 46x and 536 of volumes 4I, 4II and 5I of Fremlin's measure theory. $\endgroup$ – t.b. Aug 19 '12 at 9:13

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