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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 K = R 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 in the domain. This is same as writing f(x) de 2. Can you guess what the symbol 2 LIIS RIIS 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. Ilence 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 6. Functional Calculus for Normal Operators Define the continuous functional calculus for a normal operator N on a Hilbert space H. Show that for any continuous function f : (N) → C, there exists an operator f(N) such that f(N) respects the functional properties of f (e.g. f1(N) + ƒ₂(N) = (f1 + 2)(N)). Prove that this calculus preserves the spectrum in the sense that (f(N)) = f(0(N)). 7. Spectral Mapping Theorem for Banach Algebras Prove the spectral mapping theorem for a bounded linear operator T on a Banach space X: if p is a polynomial, then σ(p(T)) = p(o(T)). Extend this to functions defined by a holomorphic functional calculus for T, and discuss the implications of this theorem in spectral theory.

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 K = R 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 in the domain. This is same as writing f(x) de 2. Can you guess what the symbol 2 LIIS RIIS 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. Ilence 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 6. Functional Calculus for Normal Operators Define the continuous functional calculus for a normal operator N on a Hilbert space H. Show that for any continuous function f : (N) → C, there exists an operator f(N) such that f(N) respects the functional properties of f (e.g. f1(N) + ƒ₂(N) = (f1 + 2)(N)). Prove that this calculus preserves the spectrum in the sense that (f(N)) = f(0(N)). 7. Spectral Mapping Theorem for Banach Algebras Prove the spectral mapping theorem for a bounded linear operator T on a Banach space X: if p is a polynomial, then σ(p(T)) = p(o(T)). Extend this to functions defined by a holomorphic functional calculus for T, and discuss the implications of this theorem in spectral theory.

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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