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Math Help - Equicontinuity

  1. #1
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    Equicontinuity

    Let X a metric compact space and Y a Banach Space.

    Let f and g $\in$ C(X,F)

    Prove that: 1) A+B is equicontinuous, and
    2) A U B is equicontinuous

    I really dont know where to start....
    Last edited by orbit; May 15th 2011 at 08:53 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by orbit View Post
    Let X a metric compact space and Y a Banach Space.

    Let A anb B $\in$ C(X,F)

    Prove that: 1) A+B is equicontinuous, and
    2) A U B is equicontinuous

    I really dont know where to start....
    What does this mean? Equicontinuity applies to a family of maps.
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What does this mean? Equicontinuity applies to a family of maps.
    Hi, A and B are Equicontinuous subsets of C(X,F)
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  4. #4
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    Hint:

    1. |f(x_0)-f(x)+(g(x_0)-g(x))|\leq |f(x_0)-f(x)|+|g(x_0)-g(x)| < 2\varepsilon if |x-x_0|<\delta =\min \{ \delta _A, \delta _B \} whenever f\in A, \ g\in B

    2. For the union use the same \delta.
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  5. #5
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    Quote Originally Posted by Jose27 View Post
    Hint:

    1. |f(x_0)-f(x)+(g(x_0)-g(x))|\leq |f(x_0)-f(x)|+|g(x_0)-g(x)| < 2\varepsilon if |x-x_0|<\delta =\min \{ \delta _A, \delta _B \} whenever f\in A, \ g\in B

    2. For the union use the same \delta.
    Thank you.

    For 2) is the same delta for 1??

    Regars.
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  6. #6
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    I finished the exercise.

    Thank u!
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