Tensor Product: Hilbert Spaces

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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

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.

Let us choose orthonormal bases $\{e_i: i \in I\}$ of $\mathcal H$ and $\{f_j : j\in J \}$ of $\mathcal K$.

Assume $\mathcal K$ is finite dimensional, thus $J$ is finite. Let $x\in \mathcal H \hat\otimes \mathcal K$ and write it as $$ x=\sum_{j\in J} x_j \otimes f_j \quad \text{with}\quad x_j =\sum_{i\in I} (e_i\otimes e_j, x) \, e_i\,, $$ where $(\,\cdot\,,\,\cdot\,)$ is the scalar product to be chosen linear in the second argument. This shows that $x\in \mathcal H\otimes \mathcal K$.

Conversely, if both are infnite dimensional, we can identify a subset of $I$ and $J$ with $\mathbb N$. Then $$y=\sum_{n\in \mathbb N} \frac1{n!}e_n\otimes f_n \in \mathcal H\hat\otimes \mathcal K. $$ Show that $y\not \in \mathcal H\otimes \mathcal K$.