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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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) Let X be a Banach space and X* be its dual space. 1. Prove that if X is reflexive (i.e., XX**), then X is also reflexive. 2. Conversely, show that if X* is reflexive, X does not necessarily have to be reflexive. Construct an example. 3. Prove that for every finite-dimensional Banach space X, X is reflexive. Hint: Use the Hahn-Banach theorem, Alaoglu's theorem, and Krein-Milman theorem in parts of the proof.

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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) Let X be a Banach space and X* be its dual space. 1. Prove that if X is reflexive (i.e., XX**), then X is also reflexive. 2. Conversely, show that if X* is reflexive, X does not necessarily have to be reflexive. Construct an example. 3. Prove that for every finite-dimensional Banach space X, X is reflexive. Hint: Use the Hahn-Banach theorem, Alaoglu's theorem, and Krein-Milman theorem in parts of the proof.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) Let X be a Banach space and X* be its dual space. 1. Prove that if X is reflexive (i.e., XX**), then X is also reflexive. 2. Conversely, show that if X* is reflexive, X does not necessarily have to be reflexive. Construct an example. 3. Prove that for every finite-dimensional Banach space X, X is reflexive. Hint: Use the Hahn-Banach theorem, Alaoglu's theorem, and Krein-Milman theorem in parts of the proof.
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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) Let X be a Banach space and X* be its dual space. 1. Prove that if X is reflexive (i.e., XX**), then X is also reflexive. 2. Conversely, show that if X* is reflexive, X does not necessarily have to be reflexive. Construct an example. 3. Prove that for every finite-dimensional Banach space X, X is reflexive. Hint: Use the Hahn-Banach theorem, Alaoglu's theorem, and Krein-Milman theorem in parts of the proof.
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