MIT18_102s09_lec09

MIT18_102s09_lec09 - MIT OpenCourseWare http/ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 57 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 9. Thursday, March 5 My attempts to distract you all during the test did not seem to work very well. Here is what I wrote up on the board, more or less. We will use Baire’s theorem later (it is also known as ‘Baire category theory’ although it has nothing to do with categories in the modern sense). This is a theorem about complete metric spaces – it could be included in 18.100B but the main applications are in Functional Analysis. Theorem 4 (Baire) . If M is a non-empty complete metric space and C n ⊂ M, n ∈ N , are closed subsets such that [ (9.1) M = C n n then at least one of the C n ’s has an interior point. Proof. We can assume that the first set C 1 = ∅ since they cannot all be empty and dropping some empty sets does no harm. Let’s...
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This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.

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MIT18_102s09_lec09 - MIT OpenCourseWare http/ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these

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