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Homework Help: Norm in Hilbert space

  1. Oct 7, 2012 #1
    Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof.

    He states in Lemma 1.1.4:
    Let μ be a finite Borel measure on H. Then the following assertions are equivalent:
    (1) [itex] \int_H |x|^2 \mu(dx) < \infty[/itex]
    (2) There exists a positive, symmetric, trace class operator Q s.t. for [itex]x,y \in H[/itex]
    [tex] <Qx, y> = \int_H <x,z><y,z> \mu(dz).[/tex]

    If (2) holds, then [itex]Tr Q = \int_H |x|^2 \mu(dx)[/itex].
    --------

    The proof begins:
    Spse (2) holds. Let [itex](e_n)_{n\in \mathbb{N}}[/itex] be an orthonormal basis of H. Then
    [tex]
    \int_H |x|^2 \mu(dx) = \sum_{n=1}^\infty \int_H |<x, e_n>|^2 \mu(dx) = \sum_{n=1}^\infty <Qe_n, e_n> = Tr Q < \infty.
    [/tex]

    What I have trouble with is the transitiono to the sum of [itex]<Qe_n, e_n>[/itex]. If I suppose, that [itex]x, e_n[/itex] may be complex, then I miss the adjoint part of the absolute value.

    Most probably I miss some trivial notion, so any help will be appreciated.
     
    Last edited: Oct 7, 2012
  2. jcsd
  3. Oct 7, 2012 #2
    By definition, ##|z|^2 = z\overline{z} ##, so ## |<x, e_n>|^2 = <x, e_n>\overline{<x, e_n>} = <x, e_n><e_n, x> ##.
     
  4. Oct 7, 2012 #3
    I know, but what then with the following?
    [tex]
    <Qx, y> = \int_H <x,z> <y,z> \mu(dz)
    [/tex]

    If I understand correctly,
    [tex]
    <Qe_n, e_n> = \int <e_n, x> <e_n, x> \mu(dx) = \int <e_n, x>^2 \mu(dx)
    [/tex]
    which doesn't coincide with abs. value for complex numbers.
     
  5. Oct 7, 2012 #4
    Read the first paragraph in 1.1. H is a real Hilbert space.
     
  6. Oct 7, 2012 #5
    Damn, you're right! I'm deeply sorry, my trivial fault :-(
     
  7. Oct 7, 2012 #6

    dextercioby

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    Homework Helper

    I'm sorry, can you, please, post a link to the book ? I couldn't find it on google books either by name, or by title...

    Thanks! (later edit).
     
    Last edited: Oct 7, 2012
  8. Oct 7, 2012 #7
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