Topological Modules/Topological tensor products

Tensor product of Hilbert spaces

Proposition (tensor product of orthonormal bases is orthonormal basis of tensor product):

Let ${\displaystyle H_{1},H_{2}}$  be Hilbert spaces, and suppose that ${\displaystyle (e_{\lambda })_{\lambda \in \Lambda }}$  is an orthonormal basis of ${\displaystyle H_{1}}$  and ${\displaystyle (f_{\mu })_{\mu \in \mathrm {M} }}$  is an orthonormal basis of ${\displaystyle H_{2}}$ . Then ${\displaystyle (e_{\lambda }\otimes e_{\mu })_{(\lambda ,\mu )\in \Lambda \times \mathrm {M} }}$  is an orthonormal basis of ${\displaystyle H_{1}\otimes H_{2}}$ .

Proof: Let any element

${\displaystyle \sum _{j=1}^{n}f_{j}\otimes g_{j}}$

of ${\displaystyle H_{1}\otimes H_{2}}$  be given; by definition, each element of ${\displaystyle H_{1}\otimes H_{2}}$  may be approximated by such elements. Let ${\displaystyle \epsilon >0}$ . Then by definition of an orthonormal basis, we find ${\displaystyle m_{j},l_{j}}$  for ${\displaystyle j\in [n]}$  and ${\displaystyle \alpha _{j,k},\beta _{j,k},\lambda _{j,k},\mu _{j,k}}$  for ${\displaystyle j\in [n]}$  and then ${\displaystyle k\in [m_{j}]}$  resp. ${\displaystyle [l_{j}]}$  such that

${\displaystyle \left\|\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}-f_{j}\right\|<\epsilon }$  and ${\displaystyle \left\|\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}-g_{j}\right\|<\epsilon }$ .

Then note that by the triangle inequality,

${\displaystyle \left\|\sum _{j=1}^{n}f_{j}\otimes g_{j}-\sum _{j=1}^{n}\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|\leq \sum _{j=1}^{n}\left\|f_{j}\otimes g_{j}-\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|}$ .

Now fix ${\displaystyle j\in [n]}$ . 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}}}

In total, we obtain that

${\displaystyle \left\|\sum _{j=1}^{n}f_{j}\otimes g_{j}-\sum _{j=1}^{n}\left(\sum _{k=1}^{m_{j}}\alpha _{j,k}e_{\lambda _{j,k}}\right)\otimes \left(\sum _{k=1}^{l_{j}}\beta _{j,k}f_{\mu _{j,k}}\right)\right\|\leq \epsilon \sum _{j=1}^{n}\left(\|g_{j}\|+2\|f_{j}\|\right)}$

(assuming that the given sum approximates ${\displaystyle f_{j}}$  well enough) which is arbitrarily small, so that the span of tensors of the form ${\displaystyle e_{\lambda }\otimes f_{\mu }}$  is dense in ${\displaystyle H_{1}\otimes H_{2}}$ . Now we claim that the basis is orthonormal. Indeed, suppose that ${\displaystyle (\lambda ,\mu )\neq (\lambda ',\mu ')}$ . Then

${\displaystyle \langle e_{\lambda }\otimes f_{\mu },e_{\lambda '}\otimes f_{\mu '}\rangle =\langle e_{\lambda },e_{\lambda '}\rangle \langle f_{\mu },f_{\mu '}\rangle =0}$ .

Similarly, the above expression evaluates to ${\displaystyle 1}$  when ${\displaystyle \lambda =\lambda '}$  and ${\displaystyle \mu =\mu '}$ . Hence, ${\displaystyle (e_{\lambda }\otimes e_{\mu })_{(\lambda ,\mu )\in \Lambda \times \mathrm {M} }}$  does constitute an orthonormal basis of ${\displaystyle H_{1}\otimes H_{2}}$ . ${\displaystyle \Box }$