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Let $A$ be a bounded subset of $C_{[a,b]}$. I want to show that the set of functions of the form $\int_a^x f(t) dt$ for $f(t)\in A$ is compact (this problem appears in Kolmogorov & Fomin's Introduction to Real Analysis, page 107). I'm pretty sure I've successfully shown that the set is both uniformly bounded and equicontinuous. If the set is also closed, then it will follow that it is compact.

How can I go about showing that this set is closed? Am I going about the original problem the completely wrong way? If so, could someone point me in the right direction please? Thank you for any help.

ET-phone-homology
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  • The set is clearly not closed. For example, $A$ could be something like the constant functions $f=c$, $0<c<1$, and then $x-a$ is in the closure of your set, but not in the set itself. –  Mar 17 '16 at 01:13
  • Ahh, I see. Thank you for the clarification. Do you happen to have any suggestions about how to solve the original problem? – ET-phone-homology Mar 17 '16 at 01:20
  • Oh, nevermind. According to http://math.stackexchange.com/questions/757093/let-m-be-a-bounded-subset-of-the-space-c-a-b-prove-that-the-set-of-all, the claim is actually false... the best we can show is relatively compact. – ET-phone-homology Mar 17 '16 at 01:32

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