Rainwater theorem, convergence of nets, initial topology 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.

  
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*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.

  
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*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 $C(K)$.

  
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*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 $K$)?
  

 A: 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.
