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Math Help - Riesz's lemma

  1. #1
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    Riesz's lemma

    With Riesz's lemma, I know that if i let X be a normed space. and Y and Z be subspaces of a normed space X then if Y is closed proper subset of Z then for every real number theta on interval (0,1) there exist a ||z||=1 s.t.
    ||z-y||>= theta

    My question:
    When Y is finite dimensional then theta can be (0,1]

    i am having trouble proofing this.
    I believe that i want to show that since Y is bounded and closed it is compact. Therefore
    their exists value v in Z-Y s.t.
    a=inf||v-y||
    since Y is compact there exists y_o in Y such that
    a=||v-y_o||

    and if this is true i could easily finish the proof from there.

    Basically my two questions are:
    How do i Show that Y is bounded. Is it automatically implied because it is finite dimensional and closed?
    If Y is compact can i assume that there exists a y_0 in Y s.t. a=||v-y_o||?
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  2. #2
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    Quote Originally Posted by macrone View Post
    With Riesz's lemma, I know that if i let X be a normed space. and Y and Z be subspaces of a normed space X then if Y is closed proper subset of Z then for every real number theta on interval (0,1) there exist a ||z||=1 s.t.
    ||z-y||>= theta

    My question:
    When Y is finite dimensional then theta can be (0,1]

    i am having trouble proofing this.
    I believe that i want to show that since Y is bounded and closed it is compact. Therefore
    their exists value v in Z-Y s.t.
    a=inf||v-y||
    since Y is compact there exists y_o in Y such that
    a=||v-y_o||

    and if this is true i could easily finish the proof from there.

    Basically my two questions are:
    How do i Show that Y is bounded. Is it automatically implied because it is finite dimensional and closed?
    If Y is compact can i assume that there exists a y_0 in Y s.t. a=||v-y_o||?
    A subspace of a vector space can never be compact because it is not bounded. What is true is that if the subspace is finite-dimensional then its closed unit ball is compact. More generally, any closed ball in a finite-dimensional space is compact.

    The proof of Riesz's lemma shows that you can take \theta = 1 provided that the distance \inf_{y\in Y}\|v-y\| is attained. It easily follows from the triangle inequality that the smallest values of \|v-y\| must occur inside the closed ball \|y\|\leqslant2. The function y\mapsto\|v-y\| is continuous, and will therefore attain its infimum on any compact set.

    Put those facts together and you have your proof.
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