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 $K$)?