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

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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

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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

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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

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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

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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

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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

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