Consider a weak topological space.
I am trying to show or explain why that for every weak neighbourhood $N$ of $\mathbf{1}$, there is a finite set of vectors such that $N$ contains every vector $\mathbf{x}$ with the condition $$|\langle \mathbf{x-1}, \mathbf{b}\rangle| < 25$$
Does this just follow from the definition of the weak topology/weak convergence?
I don't understand why you introduce the set $S$... maybe you also wanted to ask a different question.
In any case, yes, as you anticipate, the weak topology has a local sub -basis of sets $U_{x,\epsilon}=\{v\in \ell^2: |\langle v,x\rangle|<\epsilon\}$. Then the local basis consists of finite intersections of these, which are as you've indicated.