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Math Help - Norm of a sequence of functions

  1. #1
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    Norm of a sequence of functions

    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}.

    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 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}. Do I find the limit of the sequence or what do I do to compute ||f_{n}||_{1}=\int_{a}^{b} |f_{n}| and ||f_{n}||_{2}=\sqrt{\int_{a}^{b} |f_{n}|^2}?
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  2. #2
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    Hello,

    ||f_{n}||_{1}=\int_{a}^{b} |f_{n}| and ||f_{n}||_{2}=\sqrt{\int_{a}^{b} |f_{n}|^2}?
    Just calculate these.

    For example :

    \|f_n\|_1=\int_0^1 |f_n(x)| ~dx=\int_0^{1/n} |n| ~dx

    \|g_n\|_1=\int_0^1 |g_n(x)| ~dx=\int_{1/n}^{2/n} |2n-n^2x| ~dx+\int_{2/n}^1 |0| ~dx

    (check they're indeed continuous)
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