Weak topology on a pre-Hilbert Space 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.
 A: Let $\bar X$ denote the completion.
The weak topology $\sigma(X,X)$ on $X$ is strictly coarser than the weak topology $\sigma(X,\bar X)$; see [Schaefer, Topological Vector Spaces, (IV.1.2)].
However, both topologies have the same bounded subsets (which are the norm-bounded sets). 
To see this, note that the bounded sets in $\bar X$ for $\sigma(\bar X,X)$ are contained in the completions of the bounded sets in $X$, 
for $\sigma(X,X)$. Moreover the bounded sets in $\bar X$ for $\sigma(\bar X,\bar X)$ are the same as those for $\sigma(\bar X,X)$, by Schaefer (III, 3.4).
And each equicontinuous subset of $X$ for the $\sigma(X,X)$-topology (i.e., norm bounded by the uniform boundedness theorem) the topologies $\sigma(X,X)$ and $\sigma(X,\bar X)$ coincide. 
EDIT: Correction:
The finest locally convex topology on $X$ which coincides with $\sigma(X,X)$ on each norm-ball (each bounded set), is at least as fine as the topology of uniform convergence on precompact subsets, i.e., on compact subsets of $\bar X$. It might be the same.
