Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the injective and projective tensor products of locally convex spaces. The tensor product defined below is very different, and I'd like to know how it is called and where to find information about it.
Let $H$ be a Hilbert space, $\varphi: H \to V$ and $\psi: H \to W$ be continuous linear operators, and let $\{e_\alpha\}$ and $\{f_\beta\}$ be two orthonormal bases in $H$. Then formally the following identity should be true:
$$ \sum_\alpha \varphi(e_\alpha) \otimes \psi(e_\alpha) = \sum_\beta \varphi(f_\beta) \otimes \psi(f_\beta) $$
The tensor product that I'm talking about is the space of all formal sums $ \sum_\alpha v_\alpha \otimes w_\alpha $, $v_\alpha \in V, w_\alpha \in W$, for which $\varphi(e_\alpha) := v_\alpha, \psi(e_\alpha) := w_\alpha$ extends to continuous operators from Hilbert space, modulo the identity above (and, of course, the obvious relation $\sum_\alpha v_\alpha \otimes 0 + \sum_\alpha 0 \otimes w_\alpha = 0$).
The importance of this space stems from the fact that the positive elements of $V \otimes V$ (the ones representable as $\sum v_\alpha \otimes v_\alpha$) are essentially unitary equivalence classes of maps from Hilbert space to $V$, so the topic looks like it should have been studied extensively 50 years ago.
What I know so far are just explicit descriptions of this space in several cases ($\ell^2 \otimes \ell^2$, $C \otimes C$ (RKHS theory), $\ell^1 \otimes \ell^1$ (Grothendieck), $L^0 \otimes L^0$, nuclear spaces). What should I google to find a general theory?