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(a) Show that |||| > |y(a)| d(a, Ker 4)

(a) Show that |||| > |y(a)| d(a, Ker 4)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section: Chapter Questions
Problem 5RQ
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part 2 a funct anlysis
Problem 1. Let E be a real vector space, F be a closed subspace of E, and a € E\F. Set d(a, F) inf{||ax||; x = F}. 1. Show that if F ha a finite dimension then there exists xo F such that d(x, F) = ||x-xo||. 2. Let be nonzero linear continuous form on E and a € E such that p(a) 0. We consider F Ker p. (a) Show that ||||d(a, Ker 4) |p(a)| (b) Show that (u+ta)| ||u + ta|| Ker +Ra). (c) Deduce that |||| = < |y(a)| d(a, kerp)' |p(a)| d(a, Ker ) Vu € Ker p. (Hint: Recall that E= (d) Show that if uo Ker such that d(a, Ker p) = ||a - roll then there exists To E such that|ro|| = 1 and |(ro)| = ||||- (e) Let zo E so that oto+ta with v Ker . Suppose that ||zo|| = 1 and € (0)|||||. Show that told(a, Ker y) = 1. (f) Deduce from the previous part that if zo € E such that ||zo|| = 1 and y(x) = |||| then there exists uo € Ker such that d(a, Ker y) = ||a - xo||- (g) Consider E= C([0, 1]) induced with the uniform norm and consider the line ar form: ER defined by 9($) = [*¹ f(t)dt = ['f(t}dt,_ ƒ = E. i. Show that is continuous and calculate ||||. ii. Show that if y(f)|=1 with ||f|| = 1, then f(t): What do you conclude? [1, 0<t< -1, <t<1
Transcribed Image Text:Problem 1. Let E be a real vector space, F be a closed subspace of E, and a € E\F. Set d(a, F) inf{||ax||; x = F}. 1. Show that if F ha a finite dimension then there exists xo F such that d(x, F) = ||x-xo||. 2. Let be nonzero linear continuous form on E and a € E such that p(a) 0. We consider F Ker p. (a) Show that ||||d(a, Ker 4) |p(a)| (b) Show that (u+ta)| ||u + ta|| Ker +Ra). (c) Deduce that |||| = < |y(a)| d(a, kerp)' |p(a)| d(a, Ker ) Vu € Ker p. (Hint: Recall that E= (d) Show that if uo Ker such that d(a, Ker p) = ||a - roll then there exists To E such that|ro|| = 1 and |(ro)| = ||||- (e) Let zo E so that oto+ta with v Ker . Suppose that ||zo|| = 1 and € (0)|||||. Show that told(a, Ker y) = 1. (f) Deduce from the previous part that if zo € E such that ||zo|| = 1 and y(x) = |||| then there exists uo € Ker such that d(a, Ker y) = ||a - xo||- (g) Consider E= C([0, 1]) induced with the uniform norm and consider the line ar form: ER defined by 9($) = [*¹ f(t)dt = ['f(t}dt,_ ƒ = E. i. Show that is continuous and calculate ||||. ii. Show that if y(f)|=1 with ||f|| = 1, then f(t): What do you conclude? [1, 0<t< -1, <t<1
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