Proof: Let any element
![{\displaystyle \sum _{j=1}^{n}f_{j}\otimes g_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83445994bddd7d28ab403e0fcd7e490828818306)
of
be given; by definition, each element of
may be approximated by such elements. Let
. Then by definition of an orthonormal basis, we find
for
and
for
and then
resp.
such that
and
.
Then note that by the triangle inequality,
.
Now fix
. Then by the triangle inequality,
![{\displaystyle {\begin{aligned}\left\|f_{j}\otimes g_{j}-\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|&\leq \left\|f_{j}\otimes g_{j}-\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes g_{j}\right\|+\left\|\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes g_{j}-\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|\\&=\left\|f_{j}-\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\right\|\|g_{j}\|+\left\|\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right\|\left\|g_{j}-\left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|\\&\leq \epsilon \left(\|g_{j}\|+\left\|\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right\|\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/950d6185ac999c3d5d9c7250abd0e133dc85815a)
In total, we obtain that
![{\displaystyle \left\|\sum _{j=1}^{n}f_{j}\otimes g_{j}-\sum _{j=1}^{n}\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|\leq \epsilon \sum _{j=1}^{n}\left(\|g_{j}\|+2\|f_{j}\|\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cfba1eee7fc15dd57a3d3d19d1cc0b48a182807)
(assuming that the given sum approximates
well enough) which is arbitrarily small, so that the span of tensors of the form
is dense in
.
Now we claim that the basis is orthonormal. Indeed, suppose that
. Then
.
Similarly, the above expression evaluates to
when
and
. Hence,
does constitute an orthonormal basis of
.