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Math Help - Finding subspace/continuous function

  1. #1
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    Finding subspace/continuous function

    Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

    I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?
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    Re: Finding subspace/continuous function

    Quote Originally Posted by steph3824 View Post
    Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

    I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?
    You have to prove the subspace axioms.

    U is nonempty since f\in U

    Let f,g\in U

    (f+g)(1/2)=f(1/2)+g(1/2)=f(1)+g(1)=(f+g)(1)

    How about the last axiom now.
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  3. #3
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    Re: Finding subspace/continuous function

    Would it be (af)(1/2) = af(1/2) = af(1)?
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    Re: Finding subspace/continuous function

    Quote Originally Posted by steph3824 View Post
    Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

    I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?
    I don't suppose you know the fact that if T:V\to W is a linear equation, then \ker T is a subspace of V. Well, I'm sure you can note that C[0,1]\to\mathbb{R}:f\mapsto f(1)-f(\tfrac{1}{2}) is a linear transformation and \ker f=U.
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    Re: Finding subspace/continuous function

    Quote Originally Posted by steph3824 View Post
    Would it be (af)(1/2) = af(1/2) = af(1)?
    Good.
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