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This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where elements in $D_n$ should be integrable functions defined on R.

There is a discussion on mathoverflow which says we could replace unit ball part in James theorem by convex closed set . As suggested in the comments , possibly this is needed for the question.

Could anyone help me with this ?

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    How do you show that $D_{n+1}\subset D_n$? Actually, if we wouldn't be able to find such a sequence, a theorem of James would give us that $L^1$ is reflexive (which is not true). So I think finding such a sequence will use this fact.2012-10-23
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    What do you mean by contradiction? If $\phi$ is a linear functional which doesn't take its norm (i.e. we can't find a $f$ in the closed unit ball such that $\phi(f)=\lVert\phi\rVert)$, then define $C_n:=\{f\in L^1,\lVert f\rVert\leq 1,\phi(f)\geq\lVert \phi\rVert-n^{-1}\}$. So the problem is to find such a linear functional.2012-10-23
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    Closed in what topology?2012-10-23
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    @NateEldredge At least, as the set are convex there can be closed either in the weak topology or the strong one. Do you have an other topology in mind?2012-10-23
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    @DavideGiraudo: Well, it doesn't actually say we are working in $L^1$. We could be using the uniform topology, or the product topology...2012-10-23
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    @NateEldredge Yes, at least the space on which we are working has to be clarified (I assumed it was $L^1$, but indeed, we aren't sure about that).2012-10-24
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    @DavideGiraudo I've edited to let the space be L12012-10-24

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