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])). 3. Show that if T is surjective, then there exists a constant c> 0 such that ||T*(v)|| > c|||| for all Y*.

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])). 3. Show that if T is surjective, then there exists a constant c> 0 such that ||T(v)|| > c|||| for all Y.

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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