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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KR or KC. We use the symbol, for example, f(x)=2 to say that the function f is defined by setting f(x)=r2 for all z in the domain. This is same as writing f(x) def 2. Can you guess what the symbol a2 LIIS RIS means that RIIS is defined by LIIS. 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 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 Problem 10: Hahn-Banach Theorem and Duality Let X be a normed space and Y CX a subspace. 1. Prove the Hahn-Banach theorem in full generality: if fEY, then there exists an extension eX' such that ||=|f|- 2. Use the Hahn-Banach theorem to show that every element of the dual space X* separates points in X. 3. Apply the Hahn-Banach theorem to prove that in any normed space X, the closed unit ball in X* is weak-* compact. Hint: Use dual space properties and separation arguments in each 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. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.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 KR or KC. We use the symbol, for example, f(x)=2 to say that the function f is defined by setting f(x)=r2 for all z in the domain. This is same as writing f(x) def 2. Can you guess what the symbol a2 LIIS RIS means that RIIS is defined by LIIS. 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 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 Problem 10: Hahn-Banach Theorem and Duality Let X be a normed space and Y CX a subspace. 1. Prove the Hahn-Banach theorem in full generality: if fEY, then there exists an extension eX' such that ||=|f|- 2. Use the Hahn-Banach theorem to show that every element of the dual space X* separates points in X. 3. Apply the Hahn-Banach theorem to prove that in any normed space X, the closed unit ball in X* is weak-* compact. Hint: Use dual space properties and separation arguments in each proof.

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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