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How can I prove that nested sequence of non-empty bounded closed convex sets in Hilbert space have nonempty intersection?

I just don't know where to start.


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marked as duplicate by Jonas Meyer, Mice Elf, Daniel Fischer, Ivo Terek, Conifold Aug 18 '14 at 3:17

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Do you know the weak topology for Hilbert space? –  GEdgar Nov 6 '11 at 21:03
I don't know what it is. –  Slon Nov 6 '11 at 21:18
What do you know about closed bounded convex sets in a Hilbert space, then? –  t.b. Nov 6 '11 at 21:48
Proof without weak topology is here. –  Mice Elf Aug 17 '14 at 22:15

1 Answer 1

This is Cantors Intersection Theorem. The (simple) proof can be found here for example. You need to use the fact that closed bounded convex subsets in a Hilbert space H are weakly compact.

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Unfortunately, if he doesn't know what is the weak topology, he probably also doesn't know anything about "weakly compact". This illustrates the standard wisdom: For problems with homework, seek help from the instructor. –  GEdgar Nov 7 '11 at 1:40

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