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Closed in a normed space

  1. Mar 14, 2009 #1
    Let X be a normed space, F be a closed linear subspace of X,
    Let z be in X, z is not in F.
    Let S={x+az:a is in the field Phi}=Span of F and z
    We show S is closed.

    I would define a function f : S--> Phi by f(x+az)=a
    and show that |f|< or equal to 1/d where d= distance(z,F)
    hence f is in S* (dual of S).
    So we let {x_n +a_n z} be a sequence converging to w (x_n is in F,a_n is in Phi). We show w is in S.
    We note {f(x_n +a_n z)} is a cauchy sequence.

    But I am not sure how to proceed.
     
  2. jcsd
  3. Mar 14, 2009 #2

    Dick

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    You might also note that for x_n in F, d(x_n+a_n*z,F)=d(a_n*z,F). You want to show x_n and a_n*z are both convergent separately.
     
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