In Banach spaces, the following result is well-known:
(1) Let $X$ be a Banach space. Let $\{x_n\}\subset X$ and $\{x^*_n\}\subset X^*$ be such that $x_n \rightarrow x$ (convergence with respect to strong topology on $X$) and $x^*_n\overset{\ast}{\rightharpoonup} x^*$ (convergence with respect to weak star topology on $X^*$). Then we have, $\langle x^*_n, x_n\rangle\rightarrow \langle x^*,x\rangle$.
The proof for the above result is based on the fact that if $x^*_n\overset{\ast}{\rightharpoonup} x^*$ then $\{x^*_n\}$ is bounded. We have known that this fact does not hold if we replace the sequence by net (based on the following counterexample: Must a weakly or weak-* convergent net be eventually bounded?)
My questions are:
1) Whether or not the result (1) still holds if the sequence is replaced by the net (see the following for definition: https://en.wikipedia.org/wiki/Net_(mathematics))?
2) In the case (1) is false for the net, could we construct a counterexample? And what more assumptions are added such that (1) is true for the net.
Thank you for all discussions on this topic.