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Rudin 7.24

  1. May 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Let X be a metric space, with metric d. Fix a point a in X. Assign to each p in X the function f_p defined by

    [tex]f_p(x) = d(x,p)-d(x,a)[/tex]

    where x in X.

    Prove that [itex]|f_p(x)|\leq d(a,p)[/itex] for all x in X, and that therefore, f_p in C(X).

    Prove that [itex]||f_p-f_q|| = d(p,q)[/itex] for all p,q in X.

    If [itex]\phi(p) = f_p[/itex] it follows that \phi is an isometry (a distance-preserving mapping) of X onto [itex]\phi(X) \subset C(X)[/itex].

    Let Y be the closure of \phi(X) in C(X). Show that Y is complete.
    (Conclusion: X is isometric to a dense subset of a complete metric space Y)

    Note: Rudin uses C(X) as the set of complex-valued, continuous, bounded functions with domain X.

    2. Relevant equations

    3. The attempt at a solution
    I can do everything but show that Y is complete. Of course we know that C(X) is complete. We also know that if {f_p_i} is a Cauchy sequence in \phi(X), then {p_i} is a Cauchy sequence in X.

    If {g_n} is a Cauchy sequence in Y, then
    [tex]{g_n} = {\lim_{i \to \infty} f_{n,p_i}}[/tex]
    but I don't know how to use that to prove that g_n converges.
    Last edited: May 5, 2008
  2. jcsd
  3. May 7, 2008 #2
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