Proving a set of functions is bounded in an open set

[mahler1]

Homework Statement .

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.

 

[Dick]

Homework Statement .

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.

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.

 

[mahler1]

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.

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.