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Problem
Given Hilbert spaces.
In general, their algebraic tensor product isn't complete:
$$\mathcal{H}\hat{\otimes}\mathcal{K}=\mathcal{H}\otimes\mathcal{K}\iff\dim\mathcal{H}<\infty\lor\dim\mathcal{K}<\infty$$
How to prove this from scratch?
Attempt
Choose orthonormal bases:
$$\mathcal{S}\otimes\mathcal{T}:=\{\sigma\otimes\tau:\sigma\in\mathcal{S},\tau\in\mathcal{T}\}$$
One obtains some candidates:
$$\sigma_k\otimes\tau_l\in\mathcal{S}\otimes\mathcal{T}:\quad\sum_{kl=1}^\infty\frac{1}{kl}\sigma_k\otimes\tau_l\quad\sum_{k=1}^\infty\frac{1}{kl}\delta_{kl}\sigma_k\otimes\tau_l$$
However the former one drops out:
$$\sum_{kl=1}^\infty\frac{1}{kl}\sigma_k\otimes\tau_l=\left(\sum_{k=1}^\infty\frac{1}{k}\sigma_k\right)\otimes\left(\sum_{l=1}^\infty\frac{1}{l}\tau_l\right)$$
So it is not obvious at all wether the latter one works out!
Reference
Build-up on: Vector Spaces: Tensor Product
Best Answer
That element you want to form is just an elementary tensor $x\otimes y$ in the algebraic tensor product $\mathcal H\otimes\mathcal H$. Then you want to have sums of those guys, and then limits of them.