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Consider Exercise 1.1.15 in Salamon’s Functional Analysis, we have:

Let $X$ be a compact topological space and let $Y$ be a metric space. A set $\mathscr{F}\subset C(X,Y)$ is precompact iff it is equi-continuous and pointwise precompact.

One direction is easy, on the other direction. If $\mathscr{F}$ is equi-continuous and pointwise precompact, then we let $F=\{f(x)\in Y:x\in X,f\in\mathscr{F}\}$. I want to prove that $F$ is precompact.

But I just proved that $F$ is bounded, I don’t know how to prove precompactness. Maybe I have to prove every sequence there exists a subsequence converge, but how?

Thank you for your help!

WakeUp-X.Liu
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  • Did you make an attempt to find a proof on the net or in a text book? – Kavi Rama Murthy Sep 09 '21 at 05:09
  • @KaviRamaMurthy Yes, but there are all not like this. Actually I have followed the hint to solved the all steps except this step. – WakeUp-X.Liu Sep 09 '21 at 05:31
  • Did you solve this problem？ I also met this one – zik2019 Oct 03 '21 at 13:56
  • @zik2019 Oh I haven’t solved this exercise yet. Actually I solved all the exercises in the text (what I have read so far)except this one.. So I’m going to set on it for a while and see if I can solve it.. – WakeUp-X.Liu Oct 03 '21 at 15:13
  • @DiamondVillager I have solved this exercise，but not follow its hint. I followed the proof of Folland‘s real analysis. Maybe you can refer to it. Besides，would you like to give me your email address，so I can consult you for other exercises in this book.Excuse me if it’s inconvenient. – zik2019 Oct 03 '21 at 15:25
  • @zik2019 Of course! My email is [email protected]. We can discuss some of the exercises in this book. – WakeUp-X.Liu Oct 04 '21 at 14:23

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