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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) 13. V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Rea! 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) = r² for all in the domain. ' .Fixed Point Theorem in Banach Spaces 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. Spectral Theorem for Compact Self-Adjoint Operators Question: Let T be a compact, self-adjoint operator on a Hilbert space H. Prove the spectral theorem, which asserts that T can be represented in terms of an orthonormal basis of eigenfunctions with corresponding real eigenvalues. Discuss the importance of the compactness and self-adjointness conditions in your proof. This is same as writing f(x) def. Can you guess what the symbol a²: f(x) means?, Open Mapping Theorem and Applications LIIS IS 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 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.

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 Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) 13. V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Rea! 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) = r² for all in the domain. ' .Fixed Point Theorem in Banach Spaces 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. Spectral Theorem for Compact Self-Adjoint Operators Question: Let T be a compact, self-adjoint operator on a Hilbert space H. Prove the spectral theorem, which asserts that T can be represented in terms of an orthonormal basis of eigenfunctions with corresponding real eigenvalues. Discuss the importance of the compactness and self-adjointness conditions in your proof. This is same as writing f(x) def. Can you guess what the symbol a²: f(x) means?, Open Mapping Theorem and Applications LIIS IS 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 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.

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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