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Math Help - how to modify Hahn-Banach Seperation Theorem?

  1. #1
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    how to modify Hahn-Banach Seperation Theorem?

    Suppose X is a real normed space, B is a non-empty open convex subset of X, E is a linear subspace of X, B don't intersect with E.
    Then there exists a linear functional f, such that f(x)=0 when x belong to E, and f(x)>0 when x belong to B.
    i don't know how to use Hahn-Banach Seperation Theorem to get this.Who can help me?Thx a lot.
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  2. #2
    Super Member girdav's Avatar
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    Re: how to modify Hahn-Banach Seperation Theorem?

    We can find a linear functional g such that g(x)<g(y) for all x\in B and y\in E, and since E is a linear space we have for all \alpha\in\mathbb R, g(x)<\alpha g(y), and considering \alpha positive/ negative, we get g(y)=0 for all y\in E, and g(x)<0 for all x \in B. Now consider f:=-g.
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