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Compactness in C([a,b])

  • #1

Homework Statement


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.


Homework 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.


The Attempt at a Solution


Any suggestion/hint, would be highly thanked.
 

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
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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.
 
  • #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
 

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