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Proving a set of functions is bounded in an open set

  1. Dec 5, 2013 #1
    The problem statement, all variables and given/known data.

    Let ##X## be a complete metric space and consider ##C(X)## the space of continuous functions from ##X## to ##\mathbb R## with the metric ##d_{\infty}##. Suppose that for every ##x \in X##, the set ##\{f(x): f \in C(X)\}## is bounded in ##\mathbb R##. Prove that there exist an open set ##U \subset X## and ##C>0## such that ##\forall x \in U## and ##\forall f \in C(X)##, ##|f(x)|\leq C##.


    The attempt at a solution.

    I am totally lost with this problem. I am having trouble understanding what I am trying to prove here. Would the proof of this statement mean that ##C(X)## is bounded restricted to some subset of ##X##? Can anyone suggest me where to begin? A lot of information is given in the statement: ##X## is complete, the functions are continuous, etc, but I don't know how to properly use all these facts.
     
  2. jcsd
  3. Dec 5, 2013 #2

    Dick

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    I don't get what they mean either. If C(X) is the space of all continuous functions X->R, then any constant function f(x)=M for arbitrary M in R is continous. So f(x) can be anything in R. So how can you assume f(x) is bounded for all f in C(X)? I'm not sure the question (whatever it is) has been stated correctly.
     
  4. Dec 5, 2013 #3
    You're right, it doesn't make any sense to assume what the exercise tells to assume because of what you've said. I've taken this exercise from an old exam, if I can figure out if there was any correction on the statement, I'll post it. As you've noticed, as it is now, it is incorrect.
     
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