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Hilbert Space

  1. Apr 26, 2013 #1
    1. The problem statement, all variables and given/known data
    let [itex]\ell^{2}[/itex] denote the space of sequences of real numbers [itex]\left\{a_{n}\right\}^{\infty}_{1}[/itex]

    such that

    [itex]\sum_{1 \leq n < \infty } a_{n}^{2} < \infty [/itex]

    a) Verify that [itex]\left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle = \sum_{1 \leq n < \infty } a_{n}b_{n} [/itex] is an inner product.

    b) Show that [itex]\ell^{2}[/itex] is a Hilbert Space.

    2. Relevant equations



    3. The attempt at a solution

    I did part a, I believe that was easy enough, however for part b, since we're given that

    [itex]\sum_{1 \leq n < \infty } a_{n}^{2} [/itex] = [itex]\left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{a_{n}\right\}^{\infty}_{1} \right\rangle [/itex] = [itex]\left\| \left\{a_{n}\right\}^{\infty}_{1} \right\|^{2}[/itex] < ∞

    does this mean that all sequences converge in the norm, so [itex]\ell^{2}[/itex] is complete and therefore a Hilbert Space?
     
  2. jcsd
  3. Apr 26, 2013 #2

    Dick

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    No, it's considerably more complicated than that. You need to prove that a Cauchy sequence of sequences in [itex]\ell^{2}[/itex] converges to a sequence in [itex]\ell^{2}[/itex]. I'm not an expert on this subject and if I were to try to figure out how to guide you through it, I'd probably have to look up a proof myself first. You might want to try that first. I'm kind of surprised they left this as an exercise with no other guidance.
     
  4. Apr 26, 2013 #3
    hmm any suggested readings?
     
  5. Apr 26, 2013 #4

    Dick

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