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.

  • $\begingroup$ What is $x$ here? Could you provide more background about what you are trying to do? $\endgroup$ – Cédric Travelletti May 20 at 9:38
  • $\begingroup$ I think now the question is complete.. But let me add my reasoning.. $\endgroup$ – stochastic_name_here May 21 at 9:26
  • $\begingroup$ By the Moore-Aranszjan theorem, I get that whenever $F:K\otimes K \rightarrow \mathbb{R}$ is a positive definite and symmetric operator on $K$ then there is a unique Hilbert space $H$ of functions on $K$ for which $F$ is a reproducing kernel, i.e. there exists $\phi :K \rightarrow H$, s.t. $$F(x,x')=\langle \phi(x),\phi(x')\rangle_H \; \; \; \; \; \forall x,x' \in K.$$ Now if I want to use that theorem in the setting of the question I have to fix $x\in B$: $$F_x(a_1,a_2) = \langle f(a_1\otimes a_2),x\rangle$$ but then the Hilbert space will depend on $x$. $\endgroup$ – stochastic_name_here May 21 at 9:30
  • $\begingroup$ It's not clear to me what you want to allow to depend on $x$. Something will have to depend on $x$ because otherwise you get that $\langle f(a_1 \otimes a_2), \lambda x \rangle$ is constant in $\lambda$ for some non-zero $x$ and it will follow that $f$ is the zero map. Is it just $g$ that should depend on $x$? $\endgroup$ – Rhys Steele May 21 at 10:20
  • $\begingroup$ In the construction of the Hilbert space, see e.g. the very Wikipedia page en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space, you can see that the Hilbert space H is built as the completion of the linear span of $\lbrace F(x,\cdot), \; x\in K$. So actually since our $F(a_1,a_2)=\langle f(a_1\otimes a_2),x \rangle-$ the Hilbert space formed is depending on $x\in B$. That should be the problem in my opinion. $\endgroup$ – stochastic_name_here May 21 at 11:43

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