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Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as $\langle\phi\otimes\psi,\tilde{\phi}\otimes\tilde{\psi}\rangle_0:=\langle\phi,\tilde{\phi}\rangle\langle\psi,\tilde{\psi}\rangle$ and extended by linearity.

How do I prove (best) that the resulting space is not complete?

Can the sequence $(\sum_{k=1}^N\frac{1}{\sqrt{k}}e_k)\otimes(\sum_{l=1}^N\frac{1}{\sqrt{l}}e_l)$ provide a contradiction? If so, how do I procede then?

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If the space is of finite dimension then it is complete. –  Keenan Kidwell Jun 29 at 14:24
    
Both Hilbert spaces contain copies of $\mathbb{R}$, so you can try the same trick. –  Hans Engler Jun 29 at 14:24
    
@KeenanKidwell: You're right, so I guess I need to work with contuity (that seems to become technical)... Is there a trick to do it in an elegant way? –  Freeze_S Jun 29 at 14:32
    
@Freeze_S: You need to assume that $\dim(H)$ is infinite. –  Martin Brandenburg Jun 29 at 14:59
    
@MartinBrandenburg: Yes, corrected! –  Freeze_S Jun 29 at 17:26

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