Hi I am interested in the following question.
Given some normed space $X$ with a subset $S \subset X$. If I consider $x \in \text{wcl}(S)$, where 'wcl' denotes the weak closure of $S$, then since the weak topology is not necessarily first-countable I can't state that there exists a sequence in $S$ such that $$x_{n} \rightharpoonup x~~~n \in \mathbb{N}$$ Can I alternatively state that there exists a net in $S$ such that $$x_{i} \rightharpoonup x~~~i \in I$$ If I can, then what would my directed set $I$ be defined as? Would any additional assumptions assist?
Thanks.