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{x e C([0, 1]) : x(0) = 0} with the sup norm and x(t) dt = 0. 5-8 (a) Let X = Y X: Then Y is a closed proper subspace of X. But there is no r1 E X with |a1|lo = 1 and dist(x1, Y) = 1. (Compare 5.3.) (b) Let Y be a finite dimensional proper subspace of a normed space X. Then there is some r E X with ||a|| 1 and dist(r, Y) 1. (Compare 5.3.) %3D

{x e C([0, 1]) : x(0) = 0} with the sup norm and x(t) dt = 0. 5-8 (a) Let X = Y X: Then Y is a closed proper subspace of X. But there is no r1 E X with |a1|lo = 1 and dist(x1, Y) = 1. (Compare 5.3.) (b) Let Y be a finite dimensional proper subspace of a normed space X. Then there is some r E X with ||a|| 1 and dist(r, Y) 1. (Compare 5.3.) %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 10EQ
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5-8 (a) Let X = {x € C([0, 1]) : (0) = 0} with the sup norm and 3D{=EX:/ x(t) dt = 0. Then Y is a closed proper subspace of X. But there is no r1 € X with ||x1|lo = 1 and dist(x1, Y) = 1. (Compare Y: %3D 5.3.) (b) Let Y be a finite dimensional proper subspace of a normed space X. Then there is some r E X with ||a|| 1 and dist(r, Y) = 1. (Compare %3D 5.3.)
Transcribed Image Text:5-8 (a) Let X = {x € C([0, 1]) : (0) = 0} with the sup norm and 3D{=EX:/ x(t) dt = 0. Then Y is a closed proper subspace of X. But there is no r1 € X with ||x1|lo = 1 and dist(x1, Y) = 1. (Compare Y: %3D 5.3.) (b) Let Y be a finite dimensional proper subspace of a normed space X. Then there is some r E X with ||a|| 1 and dist(r, Y) = 1. (Compare %3D 5.3.)
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