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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, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd.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 K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f is defined by setting f(x)= for all in the domain Fixed Point Theorem in Banach Spaces This is same as writing f(x) Can you guess what the symbol a² - f(x) means? LIISRIIS means that RIIS is defined by LUIS. 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 Question: Prove the Banach Fixed Point Theorem (Contraction Mapping Theorem) for a complete metric space X and a contraction mapping T : XX. Discuss applications of this theorem in solving integral and differential equations, and present an example where this theorem is crucial.

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, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd.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 K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f is defined by setting f(x)= for all in the domain Fixed Point Theorem in Banach Spaces This is same as writing f(x) Can you guess what the symbol a² - f(x) means? LIISRIIS means that RIIS is defined by LUIS. 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 Question: Prove the Banach Fixed Point Theorem (Contraction Mapping Theorem) for a complete metric space X and a contraction mapping T : XX. Discuss applications of this theorem in solving integral and differential equations, and present an example where this theorem is crucial.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 63E
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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, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.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 K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f is defined by setting f(x)= for all in the domain Fixed Point Theorem in Banach Spaces This is same as writing f(x) Can you guess what the symbol a² - f(x) means? LIISRIIS means that RIIS is defined by LUIS. 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 Question: Prove the Banach Fixed Point Theorem (Contraction Mapping Theorem) for a complete metric space X and a contraction mapping T : XX. Discuss applications of this theorem in solving integral and differential equations, and present an example where this theorem is crucial.
Transcribed Image Text: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, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.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 K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f is defined by setting f(x)= for all in the domain Fixed Point Theorem in Banach Spaces This is same as writing f(x) Can you guess what the symbol a² - f(x) means? LIISRIIS means that RIIS is defined by LUIS. 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 Question: Prove the Banach Fixed Point Theorem (Contraction Mapping Theorem) for a complete metric space X and a contraction mapping T : XX. Discuss applications of this theorem in solving integral and differential equations, and present an example where this theorem is crucial.
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