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Math Help - [SOLVED] Cauchy-Schwartz and norms

  1. #1
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    [SOLVED] Cauchy-Schwartz and norms

    Let ||-||_{1} and ||-||_{2} be the L^1-norm and the L^2-norm on C[a,b] the space of continuous real valued functions on the closed interval [a,b]: explicitly
    ||f||_{1}=\int_{a}^{b} |f|, ||f||_{2}=\sqrt{\int_{a}^{b} |f|^2}.

    Prove that ||f||_{1} \leq ||f||_{2} for all f \in C[a,b] (1). Hint: Remember the Cauchy-Schwartz inequality for integrals.

    Cauchy-Schwartz:

    \int_{a}^{b} f(x)^{2} dx \int_{a}^{b} g(x)^{2} dx \geq (\int_{a}^{b}f(x) g(x) dx)^{2} for two real integrable functions in an interval [a,b].

    Applying this to \int_{a}^{b} |f| = \int_{a}^{b} |f| \cdot 1 I get \int_{a}^{b} |f| dx \leq \sqrt{b-a} \sqrt{\int_{a}^{b} |f|^2 dx} but this is not what I want to show. Am I missing something here?
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  2. #2
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    Yes you are missing something. In lectures John Jones said that there was an error in the question and that you should replace [a,b] with [0,1].
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  3. #3
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    Oh ok, thanks!! I've been ill for 2 weeks so I've probably missed that...thanks!
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  4. #4
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    you're welcome.
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