Tensor Products of Hilbert Spaces

Given a finite or countably infinite family of Hilbert spaces \((H_j)_{j\in N} \), we study the Hilbert space tensor product \(\bigotimes_{j\in N} H_j\). In the general case, these tensor products were introduced by John von Neumann. We are especially interested in the case where each Hilbert space \(H_j\) is given as a reproducing kernel Hilbert space, i.e., \(H_j = H(K_j)\) for some reproducing kernel \(K_j\). We establish the following result, which is new for the case of N being infinite: If we restrict the domains of the kernels \(K_j\) properly, their pointwise product \(K\) is again a reproducing kernel, and \[ H(K) \cong \bigotimes_{j\in N} H_j\, \] i.e., there is an isometric isomorphism between both spaces respecting the tensor product structure