Hilbert Schmidt embedding (needed for Cylindrical Wiener processes)

Question

In Karczewska's paper (2005) it says: "Assume that $U_1$ is an arbitrary Hilbert space such that $U$ is continuously embedded into $U_1$ and the embedding of $U_0$ into $U_1$ is a Hilbert-Schmidt operator."

Q: Why does such a $U_1\supset U$ exist?

Its existence seems to be assumed, without explanation it seems, in Da Prato's and Zabczyk's book as well. In addition, the embedding $J:U_0\to U_1$ is to be thought as the inclusion map (what I deduce from reading their work). On the contrary, in the book of Prévôt and Röckner, they consider a more general embedding $J:U_0\to U_1$ and provide a proper example that such a $J$ together with $U_1$ always exists, but the example is $U=U_1$ and where $J$ is not the inclusion map.

Answer 1

Such a space exists! See An Introduction to Stochastic PDEs, version July 24, 2009, by Martin Hairer.

Answer 2

An article on this subject that I found very helpful is 'Stochastic evolution equations with a spatially homogeneous Wiener process' by Peszat and Zabczyk. They construct such a space of distributions explicitly as the dual of a Schwartz space. Having an explicit space and doing computations with it can help understand more abstract reasoning.