Lemma 4.2 Let X be a vector space over K and let (·|·) : X × X → K be a scalar product. The following holds true: (N) The mapping ||x||(--) := V (x|x) : X → R is a norm on X. (CS) |(x|y)| < ||1|l(41)||y|l(1) (x, y E X). (P) ||x + y|l?,1) = |æ|l}?1» + ||!y|/{1,) + 23R(x\y) (x, y E X ). (PI) ||r+ y|l/1,) + |x – y|l?1) 2(|||/1,) + Ily|l/1) (x, y E X ). (CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.
Lemma 4.2 Let X be a vector space over K and let (·|·) : X × X → K be a scalar product. The following holds true: (N) The mapping ||x||(--) := V (x|x) : X → R is a norm on X. (CS) |(x|y)| < ||1|l(41)||y|l(1) (x, y E X). (P) ||x + y|l?,1) = |æ|l}?1» + ||!y|/{1,) + 23R(x\y) (x, y E X ). (PI) ||r+ y|l/1,) + |x – y|l?1) 2(|||/1,) + Ily|l/1) (x, y E X ). (CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.
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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Transcribed Image Text:Lemma 4.2 Let X be a vector space over K and let (·|·) : X × X → K be a scalar
product. The following holds true:
(N) The mapping ||x||(--) := /(x|x): X → R is a norm on X.
(CS) |(x\y)| < ||l(1)||y|(-1) (x, y € X ).
(P) ||x+ y|l71-) = |x||?,!:) + ||y/|?,1,) + 2R(x|y) (x,Y E X ).
(PI) ||r + y|l&1,) + |a – yll21) = 2(|||1) + |) (, y € X).
= 2(|a||/21) + lly|l?1,) (x, y e X).
-
(CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram
identity.
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