Suppose $X$ is a Banach space and let $\{f_\alpha\}$ be a net of continuous linear functionals satisfying $\limsup_{\alpha} | f_\alpha(x) | < \infty$ for each fixed $x \in X$. Is it true that
$\sup_{\|x\| \leq 1} \limsup_{\alpha} | f_\alpha(x) | < \infty$ ?
$\limsup_\alpha \|f_\alpha\| < \infty$ ?
(Note: This question comes from a previous question of mine posted here which I removed because there was an error in the statement of the problem, pointed out by Daniel Fischer.)
I believe that if the directed set on which the net is indexed has a countable cofinal subset, then (1) can be proven by modifying the proof of PUB. I'm not sure about (2) or about the general case of (1).