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N 1. Let X := RN be the N-dimensional real vector space with the usual norm, that is, ||x|| := / x² + ··· + x²/√ for x = (x1,...,xN). For each x, y = RN, put T(x)(y) := Σ\/\±1x(k)y(k). Show that T is an isometric isomorphism from RN onto its dual space.

N 1. Let X := RN be the N-dimensional real vector space with the usual norm, that is, ||x|| := / x² + ··· + x²/√ for x = (x1,...,xN). For each x, y = RN, put T(x)(y) := Σ\/\±1x(k)y(k). Show that T is an isometric isomorphism from RN onto its dual space.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 24CM
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N 1. Let X := RN be the N-dimensional real vector space with the usual norm, that is, ||x|| := / x² + ··· + x²/√ for x = (x1,...,xN). For each x, y = RN, put T(x)(y) := Σ\/\±1x(k)y(k). Show that T is an isometric isomorphism from RN onto its dual space.
Transcribed Image Text:N 1. Let X := RN be the N-dimensional real vector space with the usual norm, that is, ||x|| := / x² + ··· + x²/√ for x = (x1,...,xN). For each x, y = RN, put T(x)(y) := Σ\/\±1x(k)y(k). Show that T is an isometric isomorphism from RN onto its dual space.
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