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Math Help - Hilbert spaces -urgent

  1. #1
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    Exclamation Hilbert spaces -urgent

    I need to show a few things and I know a bit of how to do it but I can't write it out properly
    I have to give a lecture about it in 3 hours for my honors seminar course so I need to have this pretty much perfect
    here goes

    step 1
    show that every finite dimensional pre-Hilbert space (inner product space) V with dim V = k is isomorphic to \mathbb{C}^k
    my prof told me something about orthogonal bases using the gram-schmidt process but I was never very good at linear algebra

    step 2
    using this and the knowledge that every cauchy sequence in \mathbb{C}^k is convergent show that every cauchy sequence in V is convergent
    this shows that V is complete and thus is a Hilbert space

    so I think I have the reasoning behind it all worked out but none of the actual math part of the proof

    can someone help me?
    I have a backup proof of the completeness of finite dimensional pre-hilbert spaces using induction on the dimension but it doesn't cover part 1 and doesn't give us as powerful of results



    edit:
    lecture went fine, class was 3:30 - 4:50 and at 4:45 I started stating the theorem, then the professor cut me off and proceeded to just quickly outline the proof
    so I didn't have to haha
    I still have to finish next tuesday but I don't have to go over this one :P
    Last edited by jbpellerin; January 15th 2009 at 04:06 PM.
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