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Math Help - Norms

  1. #1
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    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}.

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

    (b) For n \geq 1, define f_{n}:[0,1] \to \mathbb{R} by

    f_{n}(x)= \begin{matrix} n & if 0 \leq x \leq \frac{1}{n} \\ & \\ 2n-n^{2}x & if \frac{1}{n} \leq x \leq \frac{2}{n} \\ & \\ 0 & if \frac{2}{n} \leq x \leq 1\end{matrix}

    Caculate ||f_{n}||_{1} and ||f_{n}||_{2}.

    So the thing is that I've been ill for over a week now and couldn't go to lectures resulting in me not having lecture notes and no notion of norms.

    I have found hat Cauchy-Schwartz states that \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]. I don't really know how to use this to prove (1).

    As for (b) I am confused by the sequence of functions which I don't really know how to interpret (what is it geometrically?) or plug into ||f||_{1}=\int_{a}^{b} |f|, ||f||_{2}=\sqrt{\int_{a}^{b} |f|^2}.

    Now, if someone could just start me off that would be great!
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  2. #2
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    Krizalid's Avatar
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    for the first one.

    if f\equiv0, it's a trivial case, so suppose b>a, then \int_{a}^{b}{\left| f \right|}=\int_{a}^{b}{\left| f \right|\cdot 1}\le \sqrt{\left( \int_{a}^{b}{\left| f \right|^{2}} \right)\left( \int_{a}^{b}{1^2} \right)}, you can finish that now.
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  3. #3
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    When I do that I end up with \int_{a}^{b} |f| dx \leq \sqrt{b-a} \sqrt{\int_{a}^{b} |f|^2 dx} but this does not necessarily imply \int_{a}^{b} |f| dx \leq \sqrt{\int_{a}^{b} |f|^2 dx}, does it?
    I'm really tired so I might be missing something really obvious.
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