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])).
Linear Algebra: A Modern Introduction
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Author:David Poole
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Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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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".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad723176-4769-4b72-ab61-36dbf9d1ecb7%2F0fdf83ff-21f1-4dd4-b3a3-b19dac5f1ffb%2Fyg7arwb_processed.png&w=3840&q=75)
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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