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Hilbert, Banach and Fourier theory

  1. Feb 20, 2014 #1
    Hi. I want to get a quick overview of the theory of Hilbert spaces in order to understand fourier series and transforms at a higher level. I have a couple of questions I hoped someone could help me with.

    - First of all: Can anyone recommend any literature, notes etc.. which go through the sufficient results of Hilbert and Banach spaces (in a quick way) with the aim to understand Fourier theory?

    -What is the virtue of a Hilbert space being defined as a Banach space, i.e. a normed linear space in which any Cauchy sequence of elements converge to an element in the space?

    - How do I prove that ##L^2 (S^1)## where ##S^1## is the circle is a Banach space with the standard hilbert space norm

    $$\langle f, g\rangle = \int_{S^1} f^*(x) g(x) dx?$$
  2. jcsd
  3. Feb 20, 2014 #2


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    Last two questions:

    Since a Hilbert space is a Banach space, theorems about Banach spaces automatically apply to Hilbert space.

    For the last question, you need to prove completeness in the norm.
  4. Feb 21, 2014 #3
    What are the most important theorems we care about in Banach spaces?

    For example in QM square integrability is a very important property. What theorems guarantee square integrability?
  5. Feb 21, 2014 #4


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    Banach spaces theorems are about Banach spaces. Your question (guarantee square integrability) is something that has to be shown to justify calling a set a Hilbert space. Once that has been shown, then you can use Hilbert space theory and Banach space theory to ascertain certain properties.
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