Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Problem 3. Let X and Y be two Banach spaces and T: X→ Y be a linear continuous application. Recall that there exists a linear continuous T: Y* → X*, called the adjoint of T, such that (T)(x) = (Tr) for all & € Y* and all x € X. 1. (a) Prove that if T(X) is dense in Y, then the adjoint T*: Y* → X* is injective. (b) Prove that if T is injective, then T(X) is dense in Y. (Hint: Suppose by contradiction that T(X) is not dense in Y and use Hahn-Ban ach theorem). 2. Give an example in which T is injective but T is not surjective. (Take, e.g., X = L²([0, 1]) and Y = L¹ ([0, 1])).

Problem 3. Let X and Y be two Banach spaces and T: X→ Y be a linear continuous application. Recall that there exists a linear continuous T: Y* → X*, called the adjoint of T, such that (T)(x) = (Tr) for all & € Y* and all x € X. 1. (a) Prove that if T(X) is dense in Y, then the adjoint T*: Y* → X* is injective. (b) Prove that if T is injective, then T(X) is dense in Y. (Hint: Suppose by contradiction that T(X) is not dense in Y and use Hahn-Ban ach theorem). 2. Give an example in which T is injective but T is not surjective. (Take, e.g., X = L²([0, 1]) and Y = L¹ ([0, 1])).

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 2 adjoint operetor banach space
Problem 3. Let X and Y be two Banach spaces and T: X Y be a linear continuous application. Recall that there exists a linear continuous T*: Y* →X*, called the adjoint of T, such that (T*)(x) = (Tx) for all Y* and all x € X. 1. (a) Prove that if T(X) is dense in Y, then the adjoint T*: Y* → X* is injective. (b) Prove that if T* is injective, then T(X) is dense in Y. (Hint: Suppose by contradiction that T(X) is not dense in Y and use Hahn-Ban ach theorem). 2. Give an example in which T* is injective but T is not surjective. (Take, e.g., X = L²([0, 1]) and Y = L¹([0,1])). 3. Show that if T is surjective, then there exists a constant c> 0 such that ||T* (v)|| > c|||| for all EY".
Transcribed Image Text:Problem 3. Let X and Y be two Banach spaces and T: X Y be a linear continuous application. Recall that there exists a linear continuous T*: Y* →X*, called the adjoint of T, such that (T*)(x) = (Tx) for all Y* and all x € X. 1. (a) Prove that if T(X) is dense in Y, then the adjoint T*: Y* → X* is injective. (b) Prove that if T* is injective, then T(X) is dense in Y. (Hint: Suppose by contradiction that T(X) is not dense in Y and use Hahn-Ban ach theorem). 2. Give an example in which T* is injective but T is not surjective. (Take, e.g., X = L²([0, 1]) and Y = L¹([0,1])). 3. Show that if T is surjective, then there exists a constant c> 0 such that ||T* (v)|| > c|||| for all EY".
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps

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