Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(g) Consider E=C([0,1]) induced with the uniform norm and consider the line ar form: E→R defined by 4(f) 2(1) = [ ²³ ƒ (1)d² − [ ₁ f(t)dt- f(t)dt, fEE. 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

(g) Consider E=C([0,1]) induced with the uniform norm and consider the line ar form: E→R defined by 4(f) 2(1) = [ ²³ ƒ (1)d² − [ ₁ f(t)dt- f(t)dt, fEE. 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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
part f g ii and iii functionnal analysis
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
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS