Transcription of The Baire category theorem - UCL
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The Baire category theorem - UCL
MATHEMATICS 3103 (Functional Analysis)YEAR 2012 2013, TERM 2 HANDOUT #7: THE Baire category theorem AND ITSCONSEQUENCESWe shall begin this last section of the course by returning tothe study of general metricspaces, and proving a fairly deep result called theBaire category shall thenapply the Baire category theorem to prove three fundamentalresults in functional analysis:the Uniform Boundedness theorem , the Open Mapping theorem ,and the Closed Baire category theoremLetXbe a metric space. A subsetA Xis callednowhere denseinXif the interior ofthe closure ofAis empty, (A) = . Otherwise put,Ais nowhere dense iff it is containedin a closed set with empty interior. Passing to complements,we can say equivalently thatAis nowhere dense iff its complement contains a dense open set (why?)
As preparation for the proof of the Baire category theorem, let us prove a useful lemma due to Georg Cantor.3 Recall first that if X is a compact metric space, then any decreasing sequence F 1 ⊇ F 2 ⊇ F 3 ⊇ ... of nonempty closed sets has a nonempty intersection (why?). This is not true in general in a noncompact metric space (even one that is complete and
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