1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Compactness in C([a,b])

  1. Nov 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Let g belong to C([a,b]) be given, and let X={f belonging to C([a,b]) such that abs(f(t))<=abs(g(t)), for any t belonging to [a,b]}

    Find all g's for which X is compact.

    2. Relevant equations
    A compact set is closed and bounded. I believe that all the f's smaller tan g determine a bounded subset, as long as g is bounded. However, i dont understand so muc how can i put the closed part in. I believe that I need to work with the closure, as X will bo closed if X=cl(X), and cl(X)=all the f's smaller than g. However, im not able to put all this together, and to actually answer te question.

    3. The attempt at a solution
    Any suggestion/hint, would be highly thanked.
  2. jcsd
  3. Nov 17, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Compact doesn't mean closed and bounded (boundedness in particular is what isn't enough). We're in C[a,b] here - not some Euclidean space! So I'll let you rethink your approach. If you still need help, post back.
  4. Nov 18, 2008 #3
    you're right. quite an important concecptual mistake. i revised my notes, and well, realise that maybe with ascoli-arzela, but im lost. it seems like such a broad question...
    i would need some help if you dont mind. thank you
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook