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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 Rea! 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 S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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, Rcal and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, f(x) to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol x2f(x) means? def 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 15. Duality and Compactness in IP Spaces Let 1

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 Rea! 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 S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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, Rcal and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, f(x) to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol x2f(x) means? def 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 15. Duality and Compactness in IP Spaces Let 1

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 14E
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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 Rea! 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, Rcal and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, f(x) to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol x2f(x) means? def 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 15. Duality and Compactness in IP Spaces Let 1 <p< ∞, and let 2 be a bounded domain in IR". Consider the embedding I: W¹º(S) → LP() and prove that this embedding is compact. Then, use this result to show that the dual of W¹P(S) is isometrically isomorphic to a subspace of L4(2), where += 1. Hint: Use the Rellich-Kondrachov theorem for compact embeddings and analyze duality via Hölder's inequality. 16. Invariant Subspaces for Compact Operators Let I be a compact operator on an infinite-dimensional Banach space X. Prove that if I' is not invertible, then it has a non-trivial closed invariant subspace. Discuss whether this result applies to general operators on Banach spaces or is specific to compact operators. ⚫ Hint: Start by examining the spectral properties of T and use the structure of the range of T to construct an invariant subspace.
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 Rea! 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, Rcal and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, f(x) to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol x2f(x) means? def 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 15. Duality and Compactness in IP Spaces Let 1 <p< ∞, and let 2 be a bounded domain in IR". Consider the embedding I: W¹º(S) → LP() and prove that this embedding is compact. Then, use this result to show that the dual of W¹P(S) is isometrically isomorphic to a subspace of L4(2), where += 1. Hint: Use the Rellich-Kondrachov theorem for compact embeddings and analyze duality via Hölder's inequality. 16. Invariant Subspaces for Compact Operators Let I be a compact operator on an infinite-dimensional Banach space X. Prove that if I' is not invertible, then it has a non-trivial closed invariant subspace. Discuss whether this result applies to general operators on Banach spaces or is specific to compact operators. ⚫ Hint: Start by examining the spectral properties of T and use the structure of the range of T to construct an invariant subspace.
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