Advanced Functional Analysis Mastery Quiz Instructions: . . No partial credit will be awarded; any mistake will result in a score of 0. Submit your solution before the deadline. . Ensure your solution is detailed, and all steps are well-documented. . . No Al tools (such as ChatGPT or others) may be used to assist in solving the problems. All work must be your own. Solutions will be checked for Al usage and plagiarism. Any detected violation will result in a score of 0. Problem Let X and Y be Banach spaces, and let T: XY be a bounded linear operator. Consider the following tasks: 1. [Baire's Category Theorem and Applications] a. State and prove Baire's Category Theorem for Banach spaces. Use the theorem to prove that a complete metric space cannot be the countable union of nowhere dense sets. b. Use Baire's Category Theorem to show that if T: XY is a bounded linear operator between Banach spaces, then the set of points in X where I' is continuous is a dense G8 set. 2. [Norms and Topologies on Banach Spaces] a. Prove that a Banach space X is reflexive if and only if the canonical embedding of X into its double dual .X** is surjective. b. Let X = for 1
Advanced Functional Analysis Mastery Quiz Instructions: . . No partial credit will be awarded; any mistake will result in a score of 0. Submit your solution before the deadline. . Ensure your solution is detailed, and all steps are well-documented. . . No Al tools (such as ChatGPT or others) may be used to assist in solving the problems. All work must be your own. Solutions will be checked for Al usage and plagiarism. Any detected violation will result in a score of 0. Problem Let X and Y be Banach spaces, and let T: XY be a bounded linear operator. Consider the following tasks: 1. [Baire's Category Theorem and Applications] a. State and prove Baire's Category Theorem for Banach spaces. Use the theorem to prove that a complete metric space cannot be the countable union of nowhere dense sets. b. Use Baire's Category Theorem to show that if T: XY is a bounded linear operator between Banach spaces, then the set of points in X where I' is continuous is a dense G8 set. 2. [Norms and Topologies on Banach Spaces] a. Prove that a Banach space X is reflexive if and only if the canonical embedding of X into its double dual .X** is surjective. b. Let X = for 1
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter4: Equations Of Linear Functions
Section: Chapter Questions
Problem 9SGR
Related questions
Question
![Advanced Functional Analysis Mastery Quiz
Instructions:
.
.
No partial credit will be awarded; any mistake will result in a score of 0.
Submit your solution before the deadline.
. Ensure your solution is detailed, and all steps are well-documented.
.
.
No Al tools (such as ChatGPT or others) may be used to assist in solving the problems. All work
must be your own.
Solutions will be checked for Al usage and plagiarism. Any detected violation will result in a
score of 0.
Problem
Let X and Y be Banach spaces, and let T: XY be a bounded linear operator. Consider the
following tasks:
1. [Baire's Category Theorem and Applications] a. State and prove Baire's Category Theorem for
Banach spaces. Use the theorem to prove that a complete metric space cannot be the countable
union of nowhere dense sets.
b. Use Baire's Category Theorem to show that if T: XY is a bounded linear operator
between Banach spaces, then the set of points in X where I' is continuous is a dense G8 set.
2. [Norms and Topologies on Banach Spaces] a. Prove that a Banach space X is reflexive if and
only if the canonical embedding of X into its double dual .X** is surjective.
b. Let X = for 1<p<oo. Prove that X is reflexive, and show how the weak topology on
relates to the weak-* topology on the dual space \ell^p^*.
3. [Compact Operators and Spectral Properties] a. Prove that a compact operator T: XY
on Banach spaces satisfies the Riesz-Schauder Theorem, ie, the spectrum of T consists of a
countable set of eigenvalues with no accumulation points other than possibly zero.
b. Let T' be a compact self-adjoint operator on a Hilbert space. Prove that the point spectrum of
T. i.e., the set of eigenvalues of 'T', is a countable set that accumulates only at 0.
4. [The Open Mapping and Closed Graph Theorems] a. State and prove the Open Mapping
Theorem for Banach spaces, which states that if T: XY is a surjective continuous linear
operator between Banach spaces, then I' is an open map.
b. Prove the Closed Graph Theorem, which states that if T': XY is a linear operator
between Banach spaces and the graph of 'T' is closed, then I' is bounded.
5. [Spectral Theory of Bounded Linear Operators] a. Prove that if I' is a bounded linear operator
on a Banach space X, then the spectrum of T, denoted (T), is a closed subset of the complex
plane.
b. Let A be a normal operator on a Hilbert space H, i.e., A*AAA*. Prove that A is
diagonalizable and that the spectral theorem holds for normal operators on Hilbert spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6936b25-2d1d-401d-be91-a5691e52bd52%2F5f22b4ed-08c1-4abd-982f-86f628a7d101%2Fjwepupq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Advanced Functional Analysis Mastery Quiz
Instructions:
.
.
No partial credit will be awarded; any mistake will result in a score of 0.
Submit your solution before the deadline.
. Ensure your solution is detailed, and all steps are well-documented.
.
.
No Al tools (such as ChatGPT or others) may be used to assist in solving the problems. All work
must be your own.
Solutions will be checked for Al usage and plagiarism. Any detected violation will result in a
score of 0.
Problem
Let X and Y be Banach spaces, and let T: XY be a bounded linear operator. Consider the
following tasks:
1. [Baire's Category Theorem and Applications] a. State and prove Baire's Category Theorem for
Banach spaces. Use the theorem to prove that a complete metric space cannot be the countable
union of nowhere dense sets.
b. Use Baire's Category Theorem to show that if T: XY is a bounded linear operator
between Banach spaces, then the set of points in X where I' is continuous is a dense G8 set.
2. [Norms and Topologies on Banach Spaces] a. Prove that a Banach space X is reflexive if and
only if the canonical embedding of X into its double dual .X** is surjective.
b. Let X = for 1<p<oo. Prove that X is reflexive, and show how the weak topology on
relates to the weak-* topology on the dual space \ell^p^*.
3. [Compact Operators and Spectral Properties] a. Prove that a compact operator T: XY
on Banach spaces satisfies the Riesz-Schauder Theorem, ie, the spectrum of T consists of a
countable set of eigenvalues with no accumulation points other than possibly zero.
b. Let T' be a compact self-adjoint operator on a Hilbert space. Prove that the point spectrum of
T. i.e., the set of eigenvalues of 'T', is a countable set that accumulates only at 0.
4. [The Open Mapping and Closed Graph Theorems] a. State and prove the Open Mapping
Theorem for Banach spaces, which states that if T: XY is a surjective continuous linear
operator between Banach spaces, then I' is an open map.
b. Prove the Closed Graph Theorem, which states that if T': XY is a linear operator
between Banach spaces and the graph of 'T' is closed, then I' is bounded.
5. [Spectral Theory of Bounded Linear Operators] a. Prove that if I' is a bounded linear operator
on a Banach space X, then the spectrum of T, denoted (T), is a closed subset of the complex
plane.
b. Let A be a normal operator on a Hilbert space H, i.e., A*AAA*. Prove that A is
diagonalizable and that the spectral theorem holds for normal operators on Hilbert spaces.
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