Since there was essentially no answers on my [previous](https://mathoverflow.net/questions/197887/recontruction-of-the-weak-topolgy-from-the-scalar-product-on-a-subset-of-a-hilbe?noredirect=1#comment491399_197887) question, I will ask a partial case of it, which is very easy to state.

Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product (pre-Hilbert) space. Is it possible to describe the weak topology on $X$ explicetely in terms of $\left<\cdot,\cdot\right>$ as a function on $X\times X$? That means without mentioning such **not-enough-constructive** objects as "completion" and "the dual".

Thank you.