24.2 Lemma Let X be an inner product space and ƒ € X'. (a) Let {u₁, 2,...} be an orthonormal set in X. Then Σlf(un)1² ≤||f||². Proof: (a) For m = 1,2,..., let ym = Σf(un)un. n=1 Since {u₁,, um} is an orthonormal set, m ||ym||² = (ym, Ym) = f(un)² = 3m, say. -[if n=1 m Since f(ym) = f(un)f(un) = ßm and [ƒ(3m)| ≤ ||f|| ||ym||, we s n=1 that m≤|f||√3m, that is, ßm ≤ ||f||². Letting m→→∞ (if the set denumerable), we obtain
24.2 Lemma Let X be an inner product space and ƒ € X'. (a) Let {u₁, 2,...} be an orthonormal set in X. Then Σlf(un)1² ≤||f||². Proof: (a) For m = 1,2,..., let ym = Σf(un)un. n=1 Since {u₁,, um} is an orthonormal set, m ||ym||² = (ym, Ym) = f(un)² = 3m, say. -[if n=1 m Since f(ym) = f(un)f(un) = ßm and [ƒ(3m)| ≤ ||f|| ||ym||, we s n=1 that m≤|f||√3m, that is, ßm ≤ ||f||². Letting m→→∞ (if the set denumerable), we obtain
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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![24.2 Lemma
Let X be an inner product space and ƒ € X'.
(a) Let {u₁, ₂,...} be an orthonormal set in X. Then
Σlf(un)1² ≤||f||².
Proof:
(a) For m=1,2,..., let
Ym = Σf(un)un.
n=1
Since {u₁,, um} is an orthonormal set,
m
m
||3m||² = (ym, Ym) = |ƒ(un)|² = ßm, say.
n=1
Since f(ym) = [ƒ(un)ƒ (un) = ßm and |ƒ(ym)| ≤ ||ƒ|| ||ym||, we see
n=1
that 3m ≤||f||√m, that is, ßm ≤ ||f||². Letting m→ ∞o (if the set is
denumerable), we obtain
Σlf(un)1² ≤||f||².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15b7304-cc73-4505-92c3-23aa2fda4f71%2F0b4e47e4-a057-4aa8-820a-01fae802b5bc%2Fukqjb_processed.png&w=3840&q=75)
Transcribed Image Text:24.2 Lemma
Let X be an inner product space and ƒ € X'.
(a) Let {u₁, ₂,...} be an orthonormal set in X. Then
Σlf(un)1² ≤||f||².
Proof:
(a) For m=1,2,..., let
Ym = Σf(un)un.
n=1
Since {u₁,, um} is an orthonormal set,
m
m
||3m||² = (ym, Ym) = |ƒ(un)|² = ßm, say.
n=1
Since f(ym) = [ƒ(un)ƒ (un) = ßm and |ƒ(ym)| ≤ ||ƒ|| ||ym||, we see
n=1
that 3m ≤||f||√m, that is, ßm ≤ ||f||². Letting m→ ∞o (if the set is
denumerable), we obtain
Σlf(un)1² ≤||f||².
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