1. ## Metrics Spaces: Ball

Hello,

I have a problem with the following exercise:

Consider the metrics space (C[0,1], d1) where

d1(f,g)= $\displaystyle \int$(a,b) |f(x)-g(x)dx

Let f(x) = e^(-x) for 0<=x<=1

(i) Write the open ball B(f,1) explicitly as a set using {,} and $\displaystyle \int$.

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}

(ii) Which of the following fnctions in C[o,1] are in B(f,1)?
(a) g1(x)=0;
(b) g2(x)=2e^(-x)
(c) g3(x)=1

Are any of these functions in B(f,1/2)? Justify your answers by explicit calcultions.
For the second question, I m don't know the way how can I prove that?

Can someone help me?

Thank you

Richard

2. Originally Posted by rickgoz
Hello,

I have a problem with the following exercise:

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
.

3. Thank you very much,

I understood now how I have to solve this second question.

And about, the first one: What I did it's ok?

Sorry for english,

I am a foreign student..

Richard

4. Originally Posted by rickgoz
Thank you very much,

I understood now how I have to solve this second question.

And about, the first one: What I did it's ok?

Sorry for english,

I am a foreign student..

Richard

What you did is fine and so is your english...so far.

Tonio