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Math Help - hilbert space

  1. #1
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    hilbert space

    Hello,

    I am considering the banach-space L^{p}(\mathbb{R}) and the functions:


    f=\chi_{[0,1[}
    g=\chi_{[1,2[}

    I have calculated the numbers \Vert f \Vert_{p}, \Vert g \Vert_{p}, \Vert f+g \Vert_{p} and \Vert f-g \Vert_{p} (they all have value 1).

    How do I, using the above numbers, conclude that L^{p}(\mathbb{R}) does not form a hilbertspace for p \neq 2 ?

    Thanks.
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  2. #2
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    The parallelogram law holds in a Hilbert space.

    And BTW they shouldn't all have the value 1.
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  3. #3
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    Ok thanks. Maybe I made a mistake in calculating the values. Let me look at it.
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  4. #4
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    norms

    Here are my values:

     \Vert f \Vert_{p}=\left( \int_{0}^{1}|1|^{p}dx \right)^{\frac{1}{p}}=1

     \Vert g \Vert_{p}=\left( \int_{1}^{2}|1|^{p}dx \right)^{\frac{1}{p}}=1

     \Vert f+g \Vert_{p}=\left( \int_{0}^{1}|1|^{p}dx +  \int_{1}^{2}|1|^{p}dx\right)^{\frac{1}{p}}=2

    Could you check the last calculation???
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  5. #5
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    Quote Originally Posted by surjective View Post
    Here are my values:

     \Vert f \Vert_{p}=\left( \int_{0}^{1}|1|^{p}dx \right)^{\frac{1}{p}}=1

     \Vert g \Vert_{p}=\left( \int_{1}^{2}|1|^{p}dx \right)^{\frac{1}{p}}=1

     \Vert f+g \Vert_{p}=\left( \int_{0}^{1}|1|^{p}dx +  \int_{1}^{2}|1|^{p}dx\right)^{\frac{1}{p}}=2

    Could you check the last calculation???
    The first two are correct. For the last one, the sum of the two integrals inside the parentheses is 2, but that should then be raised to the power 1/p. So the answer should be 2^{1/p}.
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