Since there was essentially no answers on my previous 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.

Since there was essentially no answers on my previous 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.