MIT18_102s09_lec15

MIT18_102s09_lec15 - MIT OpenCourseWare http/ocw.mit.edu...

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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 .

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93 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 15. Thursday, April 2 I recalled the basic properties of the Banach space, and algebra, of bounded operators B ( H ) on a separable Hilbert space H . In particular that it is a Banach space with respect to the norm (15.1) A = sup u H =1 Au H and that the norm satisFes (15.2) AB A b . Restatated and went through the proof again of the Theorem 13 (Open Mapping) . If A : B 1 −→ B 2 is a bounded linear operator between Banach spaces and A ( B 1 ) = B 2 , i.e. A is surjective, then it is open: (15.3) A ( O ) B 2 is open O B 1 open. Proof in Lecture 13, also the two consequences of it: If A : B 1 −→ B 2 is bounded, 1-1 and onto (so it is a bijection) then its inverse is also bounded. Secondly the closed graph theorem. All this is
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MIT18_102s09_lec15 - MIT OpenCourseWare http/ocw.mit.edu...

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