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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).

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).

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
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part 1 adjoint operator banach space hahn banach
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".
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