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) .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. E. Kreyssig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Banach-Alaoglu Theorem (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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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, f(x)= r² to say that the function f is defined by setting f(x) = r² for all This is same as writing f(x) def 2. Can you guess what the symbol 2 LIIS RIIS means that RIIS is defined by LIIS. in the domain. f(x) means?. 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 Question: Prove the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed vector space is compact in the weak-* topology. Use Tychonoff's theorem as part of your proof, and explore its applications in reflexive spaces. 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 between them. This has two advantages: (1) the students get to see the typical applications sup ||Tn(x)||< ∞ for each a X, then sup,, ||T|| < ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. of the results of functional analysis to other parts of analysis and (2) while dealing with such

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) .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. E. Kreyssig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Banach-Alaoglu Theorem (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. Limaye, 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, f(x)= r² to say that the function f is defined by setting f(x) = r² for all This is same as writing f(x) def 2. Can you guess what the symbol 2 LIIS RIIS means that RIIS is defined by LIIS. in the domain. f(x) means?. 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 Question: Prove the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed vector space is compact in the weak-* topology. Use Tychonoff's theorem as part of your proof, and explore its applications in reflexive spaces. 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 between them. This has two advantages: (1) the students get to see the typical applications sup ||Tn(x)||< ∞ for each a X, then sup,, ||T|| < ∞. Include an example that illustrates a situation where the Uniform Boundedness Principle applies. of the results of functional analysis to other parts of analysis and (2) while dealing with such

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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