# Inf norm question

1. Sep 21, 2008

### tom_rylex

1. The problem statement, all variables and given/known data
I have the solution to an inhomogeneous equation:
$$u(x) = \int_{0}^{1} g(x,t)f(t)dt$$
$$g(x) = x(1-t) , 0<x<t$$
and
$$g(x) = t(1-x), x<t<1$$

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

2. Relevant equations
I already know that
$$\| u \|_{\infty} \leq \frac{1}{8} \| f \|_{\infty}$$
and
$$\| u \|_{1} \leq \frac{1}{8} \| f \|_{1}$$

3. The attempt at a solution
I think I'd like to say that
$$sup| f | \leq 2| f |$$
for some x in f, and
$$\| u \|_{\infty} \leq 2*\| u \|_1 \leq 2*\frac{1}{8} \| f \|_1$$
Therefore
$$\| u \|_{\infty} \leq \frac{1}{4} \| f \|_1$$
Is this adequate, or do I need to say something more to complete the proof?

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