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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.)

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(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.

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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 $H\otimes_{\Bbb R}{\Bbb R}\simeq H$.

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  • $\begingroup$ Thank you! Specifically, can I say the isomorphism $\phi$ from $\mathcal{H} \otimes \mathbb{R}$ to $\mathcal{H}$ is $\phi (h \otimes a)=ah$? Also, the isomorphism $\psi$ from $\mathcal{H} \otimes \mathbb{R}^m$ is $\psi (h \otimes (x_1,x_2, \cdots ,x_m))=(h \otimes x_1,h \otimes x_2, \cdots ,h \otimes x_m)$? Could you also recommend me any reference for this? I really can't find a good reference for this. $\endgroup$ Oct 4, 2019 at 19:38
  • $\begingroup$ @withgrace1040: Sorry but I have no reference for you. I'd look into books about the mathematical foundations of quantum mechanics. $\endgroup$
    – Wuestenfux
    Oct 5, 2019 at 12:17
  • $\begingroup$ I see. Thank you for your comment! Could you specify the isomorphism maps or check if the isomorphism maps I wrote are correct? Also, the definition of the hermitian operator $A^*$ above is correct? $\endgroup$ Oct 7, 2019 at 12:43
  • $\begingroup$ Regarding your second point, I think your understanding of $A^*$ is correct, I am also trying to digest this material $\endgroup$
    – Fei Cao
    Mar 6, 2021 at 23:57

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