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Math Help - Metrics Spaces: Ball

  1. #1
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    Metrics Spaces: Ball

    Hello,

    I have a problem with the following exercise:

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

    d1(f,g)= \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 \int.
    For this question, I wrote, but I am not sure about this.

    B(f,1)={g \inC[0,1]: d1(f,g)<1}
    B(f,1)={g \inC[0,1]: \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
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  2. #2
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    Quote Originally Posted by rickgoz View Post
    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 \inC[0,1]: d1(f,g)<1}
    B(f,1)={g \inC[0,1]: \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:

    d(f,g_2)=\int\limits_0^1\left|e^{-x}-2e^{-x}\right|dx=\int\limits_0^1e^{-x}dx =-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)

    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
    .
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  3. #3
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    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
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  4. #4
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    Quote Originally Posted by rickgoz View Post
    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
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