I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the minimization problem I am trying to solve. Based on https://mathoverflow.net/a/356955/137295 I would guess that the dual is the intersection of the Reprocuding Kernel Hilbert Spaces, but I have not been succesful in proofing that. Do I miss something that tells me that I am wrong? Any hint in either direction would be greatly appreciated.
Definition of $\mathcal{B}$
A Reproducing Kernel Hilbert Space (RKHS) $H_k$ is a Hilbert Space together with a positive semi-definite function $k$ of which every function $f\in H_k$ can be written like $f(x)=\int k(y,x)f(y)dy$. This function $k$ uniquely defines the RKHS. Consider the kernel $k_\pi$ given by \begin{align*} k_\pi(x,y) := \mathbb{E}_{w\sim\pi}[\sigma(w^Tx)\sigma(w^Ty)] = \int \sigma(w^Tx)\sigma(w^Ty)d\pi(w) \end{align*} where $\pi$ is some probability distribution over $\mathbb{S}^d$ and $\sigma(x) = \max(0,x)$. If we take the union over all the possible probability distributions $\pi$, then we get \begin{align*} \mathcal{B} := \cup_\pi H_{k_\pi} \end{align*} This union is indicated by the letter $\mathcal{B}$, since it is commonly called the Barron Space.
Because of the specific choices for $k_\pi$ every function in $H_{k_\pi}$ can be written like \begin{align*} f(x) = \int a(w)\sigma(w^Tx)d\pi(w) \end{align*} for some $a$. As a result we have that the norms of $H_{k_\pi}$ and $\mathcal{B}$ are given by \begin{align*} ||f||^2_{H_{k_\pi}} &= \int |a(w)|^2d\pi(w) \\ ||f||_{\mathcal{B}} &= \inf_\pi ||f||_{H_{k_\pi}} \\ \end{align*}
Possible relevant info
The inputs $x$ are elements of a compact $U\subset\mathbb{R}^d$.
$\mathcal{B}$ is a Banach space, if we consider scalar multiplication and vector addition to act on the measure. It is unknown whether it is a Hilbert Space.
$\mathcal{B}$ is isometric to $\mathcal{M}/N$ where $M$ is space of signed Radon measures with total variation norm and $N=\{\mu \in M \;|\; x\mapsto \int\sigma(w^Tx)d\mu(w) = 0 \text{ a.e.} \}$.
$\mathcal{B}$ embeds continuously in the space of Lipschitz continuous functions. The dual of the Lipschitz continuous function is given by the bounded measures with the Wasserstein 1-metric as norm.
The norm $||f||^2_{H_{k_\pi}} = \int |a(w)|^2d\pi(w)$ can also be represented as $||f||_{H_{k_\pi}} = ||a||_{L^2(U,\pi)}$. The space $L^2(U,\pi)$ is self-dual.
