Instructions to follow: * Give original work Chatgpt means downvote, *Support your work with examples and graphs where required * Follow The references: Kreyszig, Rudin and Robert. G. Bartle. Reference Books: C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, (2000) J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://matto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let KKR or KC. We use the symbol, for example, f(x)=r2 to say that the function f is defined by setting f(x)=2 for all z in the domain. This is same as writing f(x) 2. Can you guess what the symbol a2f(x) means? LIISIIS means that RIIS is defined by LIIS. I started with the principle that a first course in functional analysis is meant first as a part of the general culture and second as an important tool for any future analyst. llence the emphasis all through had been to look at concrete spaces of function and linear maps between them. This has two advantages: (1) the students get to see the typical applications of the results of functional analysis to other parts of analysis and (2) while dealing with such 3. Spectral Theory - Spectral Decomposition of Self-Adjoint Operators Let T be a self-adjoint operator on a Hilbert space H. Prove that T has a spectral decomposition, i.e., there exists a spectral measure E on H such that T = √(T) AdE(A). Outline the construction of the spectral measure and show that E is unique. Discuss the role of the spectral theorem for bounded operators in your proof. Hint: Carefully work through the spectral theorem for bounded operators, proving key intermediate steps, and using projection-valued measures. 4. Measure Theory - Construction of Product Measures Let (X,A,) and (Y, B, v) be two o-finite measure spaces. Prove that there exists a unique product measure μx on the a-algebra AB generated by the measurable rectangles Ax B with A E A and B E B, such that (x)(A x B) = μ(A)(B) for all A € A and B = B. Hint: Use the Caratheodory Extension Theorem, and provide detailed justifications for each step, including the uniqueness and existence of the product measure.

Instructions to follow: * Give original work Chatgpt means downvote, Support your work with examples and graphs where required Follow The references: Kreyszig, Rudin and Robert. G. Bartle. Reference Books: C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, (2000) J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://matto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let KKR or KC. We use the symbol, for example, f(x)=r2 to say that the function f is defined by setting f(x)=2 for all z in the domain. This is same as writing f(x) 2. Can you guess what the symbol a2f(x) means? LIISIIS means that RIIS is defined by LIIS. I started with the principle that a first course in functional analysis is meant first as a part of the general culture and second as an important tool for any future analyst. llence the emphasis all through had been to look at concrete spaces of function and linear maps between them. This has two advantages: (1) the students get to see the typical applications of the results of functional analysis to other parts of analysis and (2) while dealing with such 3. Spectral Theory - Spectral Decomposition of Self-Adjoint Operators Let T be a self-adjoint operator on a Hilbert space H. Prove that T has a spectral decomposition, i.e., there exists a spectral measure E on H such that T = √(T) AdE(A). Outline the construction of the spectral measure and show that E is unique. Discuss the role of the spectral theorem for bounded operators in your proof. Hint: Carefully work through the spectral theorem for bounded operators, proving key intermediate steps, and using projection-valued measures. 4. Measure Theory - Construction of Product Measures Let (X,A,) and (Y, B, v) be two o-finite measure spaces. Prove that there exists a unique product measure μx on the a-algebra AB generated by the measurable rectangles Ax B with A E A and B E B, such that (x)(A x B) = μ(A)(B) for all A € A and B = B. Hint: Use the Caratheodory Extension Theorem, and provide detailed justifications for each step, including the uniqueness and existence of the product measure.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 94E

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