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smanalysis
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Homework Statement
We are asked to exhibit a measurable set E such that L^p(E) is separable for p= infinity. Also, we have to show that L^infinity(E) is not separable if E contains a nondegenerate interval.
Homework Equations
A normed linear space X is separable provided there is a countable subset that is dense in X.
In my real analysis class, we showed that L^infinity[a,b] is not separable but then my professor said that L^infinity of a countable set will be separable.
The Attempt at a Solution
Going by what my professor said, I'm thinking that L^infinity(Q) would work here, but I'm having a little trouble starting out on how to show that this is separable. Any ideas? I would really appreciate some help here. For the second part of the problem, I think it just follows from the fact that L^infinity[a,b] is not separable, but I'm struggling to figure out why. Well, wouldn't [a,b] itself consist of disjoint sets of nondegenerate intervals? Then again, maybe this question is related to something different...
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