I am really trying hard to understand how to view the tensor product of a Hilbert space with an $n$-dimensional vector space as being isomorphic to an $n$-dimensional Hibert space. The specific statements are in the following. (This is from the book titled "Hypocoercivity" written by Cedric Villani, one of the Fields medalists.)
(1) Could you help me understand how to view $\mathcal{H} \otimes \mathcal{V}$ as isomorphic to $\mathcal{H}^n$?
(2) Also, here it says the adjoint operator $A^*$. Is it defined from the Hilbert space $\mathcal{H} \otimes \mathcal{V} \to \mathcal{H}$ and $\langle Ah_1, h_2 \rangle_{\mathcal{H} \otimes \mathcal{V}} = \langle h_1, A^{*}h_2 \rangle_{\mathcal{H}} $?
I'd really appreciate it if you'd help me. Thank you.
Well, there is an isomorphism
$H\otimes_{\Bbb R}{\Bbb R}^m \simeq (H\otimes_{\Bbb R}{\Bbb R})\oplus \ldots \oplus (H\otimes_{\Bbb R}{\Bbb R})$ ($m$-times)
of $\Bbb R$-vector spaces with $H\otimes_{\Bbb R}{\Bbb R}\simeq H$.