I'm trying out a problem I was given and this is the statement:
Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup norm is compact.
I have a sneaking suspicion that this statement is false but I am unable to find a suitable counterexample. I have proven that the set of real-valued and bounded functions equipped with the sup norm is complete and I have bounded but do not have totally bounded so I believe that I must find a subset that is not totally bounded for my counterexample. My efforts thus far have not been fruitful. Does anyone have any idea how to approach this?
Best Answer
This is not true. For if it were, the bounded functions on an interval would be a finite-dimesional vector space. Clearly it is not.