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Identify this notation please

  1. Jun 10, 2009 #1
    Let [tex]H=L^2(R^n)[/tex]

    Here H is a Hilbert space, R^n is of course the nth product of the real line. what is L-squared? context is a quantum text for mathematicians.

    Later the author uses [tex]H=L^2([0,1])[/tex] in another example.
  2. jcsd
  3. Jun 10, 2009 #2


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  4. Jun 10, 2009 #3
    That makes perfect sense. Thank you!
  5. Jun 11, 2009 #4
    [tex]L^2([0,1])[/tex] means functions whose square integral from 0 to 1 converges, but what is [tex]L^2(R^n)[/tex]? Functions whose square integral from [tex]-\infty[/tex] to [tex]\infty[/tex] converges? Or any sub interval of [tex]R^n[/tex]?
  6. Jun 11, 2009 #5
    It is the integral over all of Rn. For example, if n=3, then [itex]\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y,z) dx dy dz[/itex], which is also written in more compact form as [itex]\int_{\textbf{R}^3} f(\textbf{x}) d\textbf{x}[/itex]
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