MIT18_102s09_lec11

MIT18_102s09_lec11 - MIT OpenCourseWare http/ocw.mit.edu...

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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 . 67 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 11. Thursday, 12 Mar Quite a lot of new material, but all of it in the various notes and books. So, I will keep it brief. (1) Convex sets and length minimizer The following result does not need the hypothesis of separability of the Hilbert space and allows us to prove the subsequent results – especially Riesz’ theorem – in full generality. Proposition 17. If C ⊂ H is a subset of a Hilbert space which is (a) Non-empty (b) Closed (c) Convex, in the sense that v 1 ,v 1 ∈ C implies 1 ( v 1 + v 2 ) ∈ C 2 then there exists a unique element v ∈ C closest to the origin, i.e. such that (11.1) v H = inf u ∈ C u H . Proof. By definition of inf there must exist a sequence { v n } in C such that v n → d = inf u ∈ C u H . We show that v n converges and that the limit is the point we want. The parallelogram law can be written (11.2) v n − v m 2 = 2 v n 2 + 2 v m 2 − 4 ( v n + v m ) / 2 2 . Since v n → d, given > if N is large enough then n > N implies 2 v n 2 < 2 d 2 + 2 / 2 . By convexity, ( v n + v m ) / 2 ∈ C so ( v n + v m ) / 2 2 ≥ d 2 . Combining these estimates gives (11.3) n,m > N = ⇒ u ∈ C u H ≤ 4 d 2 + 2 − 4 d 2 so { v n } is Cauchy. Since H is complete, v n v ∈ C since C is closed. → Moreover, the distance is continuous so v H = lim n →∞ v n = d....
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MIT18_102s09_lec11 - MIT OpenCourseWare http/ocw.mit.edu...

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