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Approximation theory problem: show nonexistence of best approximation

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Problem 1.8 here (Link to Google books)
    Clarification: C[0,1] are the continuous functions on the interval [0,1] and let S denote the set of points in the problem, as it is stated (can't tell if it's a S or a P in the book).

    2. Relevant equations
    Have I understood the problem correctly, if I say that one way to solve the problem would be to choose the function f such that regardless of what a in A I choose, I can always find another a=a' in A such that max|f(x)-a'(x)| is smaller than max|f(x)-a'(x)| (where the max is taken over all x in S). How do I go about choosing such a function f? What should I be thinking about? This is where I'm stuck, so I'm afraid that I can't post any attempt at a solution yet.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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