Originally Posted by

**rickgoz** Hello,

I have a problem with the following exercise:

For this question, I wrote, but I am not sure about this.

B(f,1)={g$\displaystyle \in$C[0,1]: d1(f,g)<1}

B(f,1)={g$\displaystyle \in$C[0,1]: $\displaystyle \int$|e^(-x)-g(x)|dx<1}

For the second question, I m don't know the way how can I prove that?

Evaluate the integrals! For example:

$\displaystyle d(f,g_2)=\int\limits_0^1\left|e^{-x}-2e^{-x}\right|dx=\int\limits_0^1e^{-x}dx$ $\displaystyle =-e^{-1}+e^{-0}=1-\frac{1}{e}\sim 0.632>\frac{1}{2}\Longrightarrow g_2\notin B\left(f,\frac{1}{2}\right)$

$\displaystyle d(f,g_3)=\int\limits_0^1\left|e^{-x}-1\right|dx=\int\limits_0^1(1-e^{-x})dx=1+e^{-1}-1=e^{-1}<\frac{1}{2}\Longrightarrow g_3\in B\left(f,\frac{1}{2}\right)$ , and etc.

Tonio

Can someone help me?

Thank you

Richard