Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $\kappa:M\to X$, such that $\kappa\left(M\right)$ is total in $X$ and $K\left(p,q\right)=\left<\kappa\left(q\right),\kappa\left(p\right)\right>$, for every $p,q\in M$. Then we can endow $M$ with certain topologies by pulling them from $X$ by $\kappa$. However, the strong topology of $X$ can be introduced on $M$ without this procedure, since $\|\kappa\left(p\right)-\kappa\left(q\right)\|^{2}=\left<\kappa\left(p\right)-\kappa\left(q\right),\kappa\left(p\right)-\kappa\left(q\right)\right>=\left<\kappa\left(p\right),\kappa\left(p\right)\right>-2\mathrm{Re}\left<\kappa\left(q\right),\kappa\left(p\right)\right>+\left<\kappa\left(q\right),\kappa\left(q\right)\right>=K\left(p,p\right)-2\mathrm{Re}K\left(p,q\right)+K\left(q,q\right).$
Now, my question is how to define the weak topology on $M$ inherited from $X$ without actually constructing the latter, just in terms of $K$.
Another interpretation. Suppose we have a subset $M$ of a Hilbert Space $X$, such that $M^{\bot}=0$. In fact $M$ contains all information about $X$. How to reconstruct the weak topology on $M$ if we "forget" about $X$? Note that we even can forget about the linear structure and then reconstruct it, by representing elements of $M$ as functions acting on $M$ by $x\left(y\right)=\left<x,y\right>$. At the moment I do not understand how to define the weak topology even on the algebraic direct sum of the countably many copies of $\mathbb{C}$, with the natural scalar product.
Clearly this topology has to be strong enough for $K$ (in the earlier notations) to be separately continuous. Note, that this implies the correspondence $M\ni p\to K\left(p,p\right)$ to be lower semi-continuous. However this is not a sufficient condition. Another necessary condition is the following: $K\left(p,p\right)$ has to be bounded on compact sets. This is in fact, sufficient if we know that the weak topology on $M$ is compactly generated. However, this condition is not very convenient and I do not know if it is sufficient in the general case.
