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)| < |||(-»)|M4) (x, y € X). (P) ||r+ y|l?1) = "|1)* + 2R(x|y) (x, y E X ). (PI) ||r + y|l&1) + ||æ – yll1) = 2(|~|l/21) + llv|l{1) (x, y € 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)| < |||(-»)|M4) (x, y € X). (P) ||r+ y|l?1) = "|1)* + 2R(x|y) (x, y E X ). (PI) ||r + y|l&1) + ||æ – yll1) = 2(|~|l/21) + llv|l{1) (x, y € X). (CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.1: Orthogonality In Rn

Problem 34EQ

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