# The tensor product of a Hilbert space with a finite dimensional vector space and the adjoint operator on that space

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.

$$H\otimes_{\Bbb R}{\Bbb R}^m \simeq (H\otimes_{\Bbb R}{\Bbb R})\oplus \ldots \oplus (H\otimes_{\Bbb R}{\Bbb R})$$ ($$m$$-times)
of $$\Bbb R$$-vector spaces with $$H\otimes_{\Bbb R}{\Bbb R}\simeq H$$.
• 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. Oct 4, 2019 at 19:38
• 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? Oct 7, 2019 at 12:43
• Regarding your second point, I think your understanding of $A^*$ is correct, I am also trying to digest this material Mar 6, 2021 at 23:57