Hello,

I have a problem with the following exercise:

For this question, I wrote, but I am not sure about this.Consider the metrics space (C[0,1], d1) where

d1(f,g)= (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 .

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

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

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

Can someone help me?

Thank you

Richard