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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. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KR 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 in the domain This is same as writing f(x)) def. 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 part of the general culture and second as an important tool for any future analyst. Henc the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 19. Measure Theory - Regularity of Borel Measures Let μ be a Borel measure on IR" (or any complete separable metric space). Prove that is regular, meaning that for every Borel set BCR and € > 0, there exists a compact set KCB and an open set UB such that μ(B\K)

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 Oulline, Unpublished Course Notes (available at http://mtts.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 KR 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 in the domain This is same as writing f(x)) def. 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 part of the general culture and second as an important tool for any future analyst. Henc the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 19. Measure Theory - Regularity of Borel Measures Let μ be a Borel measure on IR" (or any complete separable metric space). Prove that is regular, meaning that for every Borel set BCR and € > 0, there exists a compact set KCB and an open set UB such that μ(B\K)

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. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.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 KR 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 in the domain This is same as writing f(x)) def. 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 part of the general culture and second as an important tool for any future analyst. Henc the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 19. Measure Theory - Regularity of Borel Measures Let μ be a Borel measure on IR" (or any complete separable metric space). Prove that is regular, meaning that for every Borel set BCR and € > 0, there exists a compact set KCB and an open set UB such that μ(B\K)<e and µ(U \ B) <. Hint: Carefully construct approximations to Borel sets using compact and open sets, and use inner and outer regularity. 20. Operator Theory - The Hahn-Banach Extension Theorem Prove the Hahn-Banach Extension Theorem, which states that if p: XR is a sublinear functional on a real vector space X and f: M→ R is a linear functional defined on a subspace MCX such that f(x) p(x) for all a Є M, then ƒ can be extended to a linear functional F:X→ R such that F(a)p(x) for all EX.
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. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.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 KR 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 in the domain This is same as writing f(x)) def. 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 part of the general culture and second as an important tool for any future analyst. Henc the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 19. Measure Theory - Regularity of Borel Measures Let μ be a Borel measure on IR" (or any complete separable metric space). Prove that is regular, meaning that for every Borel set BCR and € > 0, there exists a compact set KCB and an open set UB such that μ(B\K)<e and µ(U \ B) <. Hint: Carefully construct approximations to Borel sets using compact and open sets, and use inner and outer regularity. 20. Operator Theory - The Hahn-Banach Extension Theorem Prove the Hahn-Banach Extension Theorem, which states that if p: XR is a sublinear functional on a real vector space X and f: M→ R is a linear functional defined on a subspace MCX such that f(x) p(x) for all a Є M, then ƒ can be extended to a linear functional F:X→ R such that F(a)p(x) for all EX.
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