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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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org. in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x)=r2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 13. Measure Theory - Egorov's Theorem Prove Egorov's Theorem, which states that if {f} is a sequence of measurable functions on a measure space (X, F,) converging pointwise to a function f almost everywhere on X, then for every € > 0, there exists a measurable set ECX with μ(E)

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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org. in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x)=r2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 13. Measure Theory - Egorov's Theorem Prove Egorov's Theorem, which states that if {f} is a sequence of measurable functions on a measure space (X, F,) converging pointwise to a function f almost everywhere on X, then for every € > 0, there exists a measurable set ECX with μ(E)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 58E
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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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org. in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x)=r2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 13. Measure Theory - Egorov's Theorem Prove Egorov's Theorem, which states that if {f} is a sequence of measurable functions on a measure space (X, F,) converging pointwise to a function f almost everywhere on X, then for every € > 0, there exists a measurable set ECX with μ(E)<e such that {f} converges uniformly to fon X E Hint: Use properties of almost everywhere convergence, constructing E as the set where convergence is slower than required for uniformity. 14. Operator Theory - The Polar Decomposition of Bounded Operators Let T be a bounded operator on a Hilbert space H. Prove the Polar Decomposition theorem, which states that there exists a partial isometry U on H and a positive semi-definite operator |T|= (TT) such that T=UT. Show that U is unique on the orthogonal complement of the kernel of T.
Transcribed Image Text: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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org. in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x)=r2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 13. Measure Theory - Egorov's Theorem Prove Egorov's Theorem, which states that if {f} is a sequence of measurable functions on a measure space (X, F,) converging pointwise to a function f almost everywhere on X, then for every € > 0, there exists a measurable set ECX with μ(E)<e such that {f} converges uniformly to fon X E Hint: Use properties of almost everywhere convergence, constructing E as the set where convergence is slower than required for uniformity. 14. Operator Theory - The Polar Decomposition of Bounded Operators Let T be a bounded operator on a Hilbert space H. Prove the Polar Decomposition theorem, which states that there exists a partial isometry U on H and a positive semi-definite operator |T|= (TT) such that T=UT. Show that U is unique on the orthogonal complement of the kernel of T.
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