MIT18_102s09_lec17

MIT18_102s09_lec17 - 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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100 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 17. Thursday April 9 was the second test (1) Problem 1 Let H be a separable (partly because that is mostly what I have been talking about) Hilbert space with inner product ( ) and norm · . · , · Say that a sequence u n in H converges weakly if ( u n ,v ) is Cauchy in C for each v H. (a) Explain why the sequence u n H is bounded. Solution: Each u n deFnes a continuous linear functional on H by (17.1) T n ( v ) = ( v,u n ) , T n = u n ,T n : H −→ C . ±or Fxed v the sequence T n ( v ) is Cauchy, and hence bounded, in C so by the ‘Uniform Boundedness Principle’ the T n are bounded, hence u n is bounded in R . (b) Show that there exists an element u H such that ( u n ,v ) ( u,v ) for each v H. Solution: Since ( v,u n ) is Cauchy in C for each Fxed v H it is convergent. Set (17.2) Tv = lim ( v,u n ) in C . n →∞ This is a linear map, since (17.3) T ( c 1 v 1 + c 2 v 2 ) = lim c 1 ( v 1 ,u n ) + c 2 ( v 2 ,u ) = c 1 Tv 1 + c 2 Tv 2 n →∞ and is bounded since | Tv | ≤ C v , C = sup n u n . Thus, by Riesz’ theorem there exists u H such that Tv = ( v,u ) . Then, by deFnition of T, (17.4) ( u n ,v ) ( u,v ) v H. (c) If e i , i N , is an orthonormal sequence, give, with justiFcation, an example of a sequence u n which is not weakly convergent in H but is such that ( u n ,e j ) converges for each j. Solution:
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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_lec17 - MIT OpenCourseWare http:/ocw.mit.edu...

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