I'd like to have a reference (book) for the following kind of definition regarding tensor products of Hilbert spaces:
Definition. Let $H_1$, $H_2$ denote complex, separable Hilbert spaces with inner products denoted by $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$, respectively. We call a pair $(H,\otimes)$, where $H$ is a Hilbert space with inner product $\langle \cdot,\cdot\rangle_H$ and $\otimes:H_1\times H_2\longrightarrow H$ a bilinear map, (a realization of) the tensor product of $H_1$ and $H_2$ if it holds that
$$\langle x_1\otimes x_2,y_1\otimes y_2\rangle_H=\langle x_1,y_1\rangle_1 \langle x_2\otimes y_2\rangle_2$$
and if $\{e_n\}_{n\in \mathbb N}$ and $\{f_k\}_{k\in \mathbb N}$ are orthonormal bases in $H_1$ and $H_2$, respectively, then $\{e_n\otimes f_k\}_{n,k}$ is an orthonormal basis in $H$.
I've found definitions which replace the second condition with the requirement that the set $\{x\otimes y\}$ should be total in $H$, i.e. its linear span must be dense. I think that both definitions are equivalent. However, I'd like to have a reference for the statement above, which I've only seen in some lecture notes.
Best Answer
I'm afraid I don't have a reference at hand. However, I can sketch out a proof that the two definitions are equivalent.
The hard direction of implication is showing that if $\text{Span}(\{ x \otimes y : x \in H_1, y \in H_2 \})$ is dense in $H$, then $\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$ is dense in $H$.
Step 1: Show that for any $x \in H_1$ and $y \in H_2$, there exists a sequence $(z_n)_{n \in \mathbb N}$ with terms in $\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$, such that $\lim_{n \to \infty} z_n = x \otimes y$.
It easy to see how to construct $z_n$.
Now define $ z_n := x_n \otimes y_n$, for each $n \in \mathbb N$.
It's clear that $z_n \in \text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$. With a little work, you can verify that $\lim_{n \to \infty} z_n = x \otimes y$.
[The technique here is to write \begin{align}\left\| z_n - x \otimes y \right\|_H^2 & \leq \left\| z_n - x \otimes y_n \right\|_H^2 + \left\| x \otimes y_n - x \otimes y \right\|_H^2 \\ & = \left\| x_n - x\right\|_{H_1}^2 \left\| y_n \right\|_{H_2}^2 + \left\| x\right\|_{H_1}^2 \left\| y_n - y \right\|_{H_2}^2 \end{align} and then show that the limit of the right hand side as $n \to \infty$ is zero.]
Step 2: Deduce that for any $z \in \text{Span}(\{ x \otimes y : x \in H_1, y \in H_2 \})$, there exists a sequence $(z_n)_{n \in \mathbb N}$ with terms in $\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$, such that $\lim_{n \to \infty} z_n = z$.
This follows almost immediately from the result of Step 1.
Step 3: Conclude that $\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$ is dense in $H$.
From Step 2, we know that $$ \text{Span}(\{ x \otimes y : x \in H_1, y \in H_2 \}) \subset \overline{\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})} \subset H.$$
Hence $$ H = \overline{\text{Span}(\{ x \otimes y : x \in H_1, y \in H_2 \})} \subset \overline{\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})} \subset H. $$
This shows that $\text{Span}(\{e_k \otimes f_l : k, l \in \mathbb N \})$ is dense in $H$.
The set $\{e_k \otimes f_l : k, l \in \mathbb N \}$ is clearly orthonomal, and we've just shown that its span is dense in $H$. So $\{e_k \otimes f_l : k, l \in \mathbb N \}$ is an orthonomal basis for $H$.