Let $H, K$ be two Hilbert spaces with orthonormal bases $\{e_{\alpha}\}, \{f_{\beta}\}$, respectively. Now, I already found a result proving that a basis for the Fermi Fock space generated from $H$, denoted as $\mathscr{F}_-(H)$ (i.e. the anti-symmetrised Fock space), is given by $$a^{\dagger}_H(e_{\alpha_1}) \cdots a^{\dagger}_H(e_{\alpha_1}) \Omega_H, $$ when $\{e_{\alpha_1}, \ldots, e_{\alpha_n}\}$ runs over finite subsets of $\{e_{\alpha}\}$. In the above, $a^{\dagger}_H$ and $\Omega_H$ are the creation operators and vacuum vector on $\mathscr{F}_-(H)$, respectively. Now, I want to build an orthonormal basis for the space $\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$. My first step to this was to formulate creation/annihilation operators on $\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$ that obey the CAR relations. These are given by
$$b^{\dagger}(h,k) := \frac{1}{\sqrt{2}}\left(a^{\dagger}_H(h) \otimes 1_K + \left(1_H\right)^N\otimes a_K^{\dagger}(k)\right),$$
where $h\in H, k\in K$ and $N$ is the particle number operator on $H$ and $1_H, 1_K$ are the identities on the subscripted spaces. I can show that all vectors of the form $$b^{\dagger}(e_{\alpha_1}, f_{\beta_1}) \cdots b^{\dagger}(e_{\alpha_n}, f_{\beta_n})\Omega_H\otimes \Omega_K$$ are orthonormal.$$$$ What I want to prove now is that the span of these vectors is dense in $\mathscr{F}_-(H) \otimes \mathscr{F}_-(K)$ when $\{ (e_{\alpha_1}, f_{\beta_1}, \ldots, (e_{\alpha_n}, f_{\beta_n})\}$ runs over finite subsets of $\{(e_{\alpha}, f_{\beta})\}$. To me it makes sense that this should be the case since the set is built from the creation operators on the space, but I'm struggling to rigorously prove it.