# 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 | 2 Comments

## 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

## 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

## 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

## 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