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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Request Explain underlined portion from "Functional Analysis" by BV Limaye

22.6 Lemma Let {a} be an orthonormal set in an inner product space X and TEX. Let Ez = {ua (t, ua) #0}. Then E is a countable set, say E₂ = {₁, 2,...}, and ((x, un)) { l². If E, is denumerable, then (x, un) → 0 as n - →∞. Further, if X is a Hilbert space, then Σn (x, un)un converges in X to some y such that (z-y) ua, that is, (2, ua) (y, ua) for every a. Proof: If x = 0, there is nothing to prove. Let x 0. For j = 1,2,..., let Ej = {ua |||| ≤j|(t, ua)|}. Fix j. Suppose that E, contains distinct elements Uaam. Then mj 0<m₂||x||² <j²|(x, uan)1². sjªle, Request explain how
Transcribed Image Text:22.6 Lemma Let {a} be an orthonormal set in an inner product space X and TEX. Let Ez = {ua (t, ua) #0}. Then E is a countable set, say E₂ = {₁, 2,...}, and ((x, un)) { l². If E, is denumerable, then (x, un) → 0 as n - →∞. Further, if X is a Hilbert space, then Σn (x, un)un converges in X to some y such that (z-y) ua, that is, (2, ua) (y, ua) for every a. Proof: If x = 0, there is nothing to prove. Let x 0. For j = 1,2,..., let Ej = {ua |||| ≤j|(t, ua)|}. Fix j. Suppose that E, contains distinct elements Uaam. Then mj 0<m₂||x||² <j²|(x, uan)1². sjªle, Request explain how
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