Let $B$ be a Banach space. Let $B^*$ be its dual, and let $K\subseteq B^*$ be some linear dense subspace. Denote the duality pairing between $B$ and $B^*$ via $\langle\cdot ,\cdot \rangle$. Suppose we are given a symmetric and linear map $f:K\otimes K \rightarrow B^*$. Does there exists an Hilbert space $H$ with inner product $[\cdot, \cdot]_H$ and a linear and bounded map $g:B^*\rightarrow H$ such that
$$[ga_1,ga_2]_H =\langle f(a_1 \otimes a_2),x \rangle \; \; \; \; \; \forall a_1,a_2 \in K \; \; \; \; x\in B $$
This kind of question led me to thinking about the reproducing Kernel Hilbert space. But maybe this way could not be correct.