MIT18_102s09_lec26 - MIT OpenCourseWare http:/

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MIT OpenCourseWare 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: .
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147 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 26. Thursday, May 14:Review Now, there was one fnal request beFore I go through a quick review oF what we have done. Namely to state and prove the Hahn-Banach Theorem. This is about extension oF Functionals. Stately starkly, the basic question is: Does a normed space have any non-trivial continuous linear Functionals on it? That is, is the dual space always non-trivial (oF course there is always the zero linear Functional but that is not very amusing). We did not really encounter this problem since For a Hilbert space, or even a pre-Hilbert space, there is always the space itsefl, giving continuous linear Functionals through the pairing Riesz’ Theorem says that in the case oF a Hilbert space that is all there is. I could have used the Hahn-Banach Theorem to show that any normed space has a completion, but I gave a more direct argument For this, which was in any case much more relevant For the cases oF L 1 ( R ) and L 2 ( R ) For which we wanted concrete completions. Theorem 19 (Hahn-Banach) . If M V is a linear subspace of a normed space and u : M −→ C is a linear map such that (26.1) | u ( t ) | ≤ C t V t M then there exists a bounded linear functional U U : V −→ C with U C and = u. M ±irst, by computation, we show that we can extend any continuous linear Func- tional ‘a little bit’ without increasing the norm. Lemma 20. Suppose M V is a subspace of a normed linear space, x / M and u : M −→ C is a bounded linear functional as in (26.1) then there exists = { t V ; t = t + ax, a C such that u : M (26.2) u u ( t + ax ) C t + ax V , t M, a C . = u, | | M Proof. Note that the decompositon t = t + ax oF a point in M is unique, since t + ax
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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_lec26 - MIT OpenCourseWare http:/

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