Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Request Explain underlined portion from "Functional Analysis" by BV Limaye
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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