(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Inf norm question

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