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Homework Help: Inf norm question

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    I have the solution to an inhomogeneous equation:
    [tex]

    u(x) = \int_{0}^{1} g(x,t)f(t)dt

    [/tex]
    [tex]

    g(x) = x(1-t) , 0<x<t

    [/tex]
    and
    [tex]

    g(x) = t(1-x), x<t<1

    [/tex]

    Show that
    [tex]

    \| u \|_{\infty} \leq \frac{1}{4} \| f \|_1

    [/tex]

    2. Relevant equations
    I already know that
    [tex]

    \| u \|_{\infty} \leq \frac{1}{8} \| f \|_{\infty}

    [/tex]
    and
    [tex]

    \| u \|_{1} \leq \frac{1}{8} \| f \|_{1}

    [/tex]

    3. The attempt at a solution
    I think I'd like to say that
    [tex]

    sup| f | \leq 2| f |

    [/tex]
    for some x in f, and
    [tex]

    \| u \|_{\infty} \leq 2*\| u \|_1 \leq 2*\frac{1}{8} \| f \|_1

    [/tex]
    Therefore
    [tex]

    \| u \|_{\infty} \leq \frac{1}{4} \| f \|_1

    [/tex]
    Is this adequate, or do I need to say something more to complete the proof?
     
  2. jcsd
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