This construction is found in Methods of modern mathematical physics vol. 1. See page 49
However it seems like this construction is flawed, if $(\cdot,\cdot)$ indeed is an inner product on $\mathcal{E}$ (for which he constructs $H_1\otimes H_2$ as its completion). For any $a\in\mathbb{C}$ and $\phi\in H_1,\psi\in H_2$ $$ a\phi\otimes \psi(x,y)= (x, \overline{a}\phi_1)(y,\phi_2) = (\bar{a}\phi_1)\otimes \phi_2(x,y). $$ for any $(x,y)\in H_1\times H_2$. That is, $a(\phi\otimes \psi) = (\bar{a}\phi)\otimes \psi$.
But if $(\cdot,\cdot)$ is a inner product it is conugate linear in the second argument. Thus $$ a\langle \eta\otimes \mu , \phi \otimes \psi\rangle =\langle \eta\otimes \mu , \overline{a}(\phi \otimes \psi)\rangle=\langle \eta\otimes \mu , (a\phi) \otimes \psi)\rangle = ( \eta, a\phi)(\mu,\psi) = \overline{a}( \eta, \phi)(\mu,\psi)=\overline{a}\langle \eta\otimes \mu , \phi \otimes \psi\rangle $$ for any $\eta,\phi\in H_1$ and $\mu,\psi\in H_2$.
What am I not understanding?