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Instructions to follow: * Give original work *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 S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd.. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Uniform Boundedness Principle (Banach-Steinhaus Theorem) Question: Let {T} be a sequence of bounded linear operators from a Banach space X to a normed space Y. Prove the Uniform Boundedness Principle, and demonstrate that if sup,T(x)||< ∞ for each a Є X, then sup, ||T||< ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. Riesz Representation Theorem for Hilbert Spaces Question: Let H be a Hilbert space. Prove the Riesz Representation Theorem, which states that for every continuous linear functional f on H, there exists a unique element y EH such that f(x)=(x, y) for all H. Discuss the uniqueness and existence aspects of this representation. Throughout these notes we let We nee the sambal for exampla

Instructions to follow: * Give original work *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 S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd.. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Uniform Boundedness Principle (Banach-Steinhaus Theorem) Question: Let {T} be a sequence of bounded linear operators from a Banach space X to a normed space Y. Prove the Uniform Boundedness Principle, and demonstrate that if sup,T(x)||< ∞ for each a Є X, then sup, ||T||< ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. Riesz Representation Theorem for Hilbert Spaces Question: Let H be a Hilbert space. Prove the Riesz Representation Theorem, which states that for every continuous linear functional f on H, there exists a unique element y EH such that f(x)=(x, y) for all H. Discuss the uniqueness and existence aspects of this representation. Throughout these notes we let We nee the sambal for exampla

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 5E
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Instructions to follow: * Give original work *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://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd.. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Uniform Boundedness Principle (Banach-Steinhaus Theorem) Question: Let {T} be a sequence of bounded linear operators from a Banach space X to a normed space Y. Prove the Uniform Boundedness Principle, and demonstrate that if sup,T(x)||< ∞ for each a Є X, then sup, ||T||< ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. Riesz Representation Theorem for Hilbert Spaces Question: Let H be a Hilbert space. Prove the Riesz Representation Theorem, which states that for every continuous linear functional f on H, there exists a unique element y EH such that f(x)=(x, y) for all H. Discuss the uniqueness and existence aspects of this representation. Throughout these notes we let We nee the sambal for exampla
Transcribed Image Text:Instructions to follow: * Give original work *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://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd.. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Uniform Boundedness Principle (Banach-Steinhaus Theorem) Question: Let {T} be a sequence of bounded linear operators from a Banach space X to a normed space Y. Prove the Uniform Boundedness Principle, and demonstrate that if sup,T(x)||< ∞ for each a Є X, then sup, ||T||< ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. Riesz Representation Theorem for Hilbert Spaces Question: Let H be a Hilbert space. Prove the Riesz Representation Theorem, which states that for every continuous linear functional f on H, there exists a unique element y EH such that f(x)=(x, y) for all H. Discuss the uniqueness and existence aspects of this representation. Throughout these notes we let We nee the sambal for exampla
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