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Norms in Hilbert Spaces.

  1. Aug 16, 2009 #1

    MathematicalPhysicist

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    1. The problem statement, all variables and given/known data
    Prove that for q>=p and any f which is continuous in [a,b] then [tex]|| f ||_p<=c* || f ||_q[/tex], for some positive constant c.


    2. Relevant equations
    The norm is defined as: [tex]||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}[/tex].


    3. The attempt at a solution
    Well, I think that because f is continuous so are f^p and f^q are continuous and on a closed interval which means they get a maximum and a minimum in the interval which are both positive (cause if f were zero then the norm would be zero and the ineqaulity will be a triviality), so f^q>=M2, f^p<=M1, and we get that:
    [tex]||f||_p/||f||_q<=M1^{1/p}/M2^{1/q}(b-a)^{1/p-1/q}[/tex] which is a constant.

    QED, or not?

    Thanks in advance.
     
  2. jcsd
  3. Aug 16, 2009 #2
    Your norm is wrong. It should be

    [tex]
    \|f\|_p=\left(\int_{a}^{b} |f|^p\right)^\frac{1}{p}
    [/tex]

    This means that you need to argue using |f|, not f. Otherwise, the argument is OK.
     
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