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Monthly Archives: June 2011
On trees: Compactness and finite splitting
Terms and notation are defined here. (Kechris exercise 4.11) (a) Let be a pruned tree on . It follows that is compact iff is finite splitting. (b) In particular, if is compact, there is such that for all , for … Continue reading
Posted in Descriptive Set Theory, Kechris
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Linear extension of “linear” map on dense subset of Banach space
Let be Banach spaces over the field . Let be dense in , be dense in , such that is closed under linear combinations. Let , such that for all , , and for all , . Then, there exists … Continue reading
Posted in Analysis
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König’s Lemma (p20 (A))
(Exercise 4.12) Let be a tree on . If is finite splitting, then iff is infinite. Show that this fails if is not finite splitting. Proof. “”. Suppose is finite. Then, obviously, . “”. Suppose is infinite. Since is finite … Continue reading
Posted in Descriptive Set Theory, Kechris
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Trees
Definitions and notation for trees. . If , and , . If , the concatenation of and is defined by . If , , then is an initial segment of and is an extension of (written ), if , for … Continue reading
Posted in Definition, Descriptive Set Theory, Kechris
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A complete, compatible metric for the strong topology (p15 (A))
Let be separable Banach spaces over . Show that the following is a complete, compatible metric for the strong topology on : where is a dense sequence in the unit ball of . Proof. It is not difficult to check … Continue reading
Any subspace of a separable metric space is separable
Let be a separable, metric space, , and let be countable and dense. For each , choose if such intersection is nonempty. Then, is a countable subset of . Let . Then choose . Now, choose such that . Then, , so … Continue reading
Posted in Metric Spaces, Separability, Topology
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Weak* and product topology are the same on $B_1((\ell^1)^*) = D^N$ (p19 (C))
Let . Show that and that the weak*topology on is the same as the product topology on . Remarks Without further comment, I will interchange between and . Also, I will sometimes abbreviate to simply . A basic open neighborhood … Continue reading
Posted in Analysis, Descriptive Set Theory, Kechris
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