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Good day,

It is known that every Hilbert space is reflexive. Does the same hold true for Pre-Hilbert spaces? I guess not since completeness is a pretty strong property that is now missing and the classical proof (as a corollary by Riesz) shouldn't work anymore. What do you think?

So if they are not reflexive could you give me an example of an non-reflexive Pre-Hilbert space?

Thanks a lot, Marvin

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    $\begingroup$ No, if a pre-hilbert space is not complete it is not reflexive, dual spaces are always complete! On the other hand if a pre-hilbert space is complete, it is a hilbert space. $\endgroup$
    – s.harp
    Dec 19, 2016 at 17:30
  • $\begingroup$ @s.harp Ah, right, thanks, I forgot that every reflexive normed vector space is automatically complete. So every Pre-Hilbert space that is not complete is therefore not reflexive. $\endgroup$
    – Cahn
    Dec 19, 2016 at 17:36

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No, if a pre-Hilbert space is not complete it is not reflexive, dual spaces are always complete! On the other hand if a pre-Hilbert space is complete, it is a Hilbert space.

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