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Thread: Norm of a sequence of functions

  1. #1
    Oct 2008

    Norm of a sequence of functions

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

    For $\displaystyle n \geq 1$, define $\displaystyle f_{n}:[0,1] \to \mathbb{R} $ by

    $\displaystyle 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 $\displaystyle ||f_{n}||_{1}$ and $\displaystyle ||f_{n}||_{2}$.

    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 $\displaystyle ||f||_{1}=\int_{a}^{b} |f|$, $\displaystyle ||f||_{2}=\sqrt{\int_{a}^{b} |f|^2}$. Do I find the limit of the sequence or what do I do to compute $\displaystyle ||f_{n}||_{1}=\int_{a}^{b} |f_{n}|$ and $\displaystyle ||f_{n}||_{2}=\sqrt{\int_{a}^{b} |f_{n}|^2}$?
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  2. #2
    Moo is offline
    A Cute Angle Moo's Avatar
    Mar 2008
    P(I'm here)=1/3, P(I'm there)=t+1/3

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

    For example :

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

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

    (check they're indeed continuous)
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