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Please, I need a small help with a reference.

Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and bounded.

I have heard, that the generalized cantor's theorem is the way to prove the existence, but I am unfortunately unable to find the proof, or the theorem itself (other than the set theory with $2^A$ $2^{2^A}$...).

Do You know about any reference, that would supply me with the proof?

Thank You very much!

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Found it out.

There might have been a mistake, the proof should not be Cantors generalized theorem, but just a generalization of the basic fact, that a continuous function on a compact closed set attains extremas.

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    $\begingroup$ Hi, since you asked for a reference - could you add the reference or a proof here, so other people can learn as well? $\endgroup$
    – Amir Sagiv
    Commented Apr 15, 2016 at 8:48
  • $\begingroup$ I would love to, but the reference was verbal - "it is just the continuous function on a compact closed set, man" ... so if somebody would suggest a reference with a proof, I will accpt his solution rather (this answer was 1 week without a notice so I added a .. mmm semi-answer...) $\endgroup$
    – Holi
    Commented Apr 16, 2016 at 17:52

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