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 S. Kumaresan, Real Analysis - An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 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. This is same as writing f(x) de 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 Stone-Weierstrass Theorem in Spectral Theory Let C(X) denote the space of continuous complex-valued functions on a compact Hausdorff space X. Prove the Stone-Weierstrass theorem, which states that any subalgebra of C(X) that separates points, contains the constant functions, and is closed under complex conjugation, is dense in C(X) under the sup norm. Then, discuss how this theorem can be applied in spectral theory to approximate operators. Unbounded Operators and the Spectrum Define the spectrum of an unbounded operator T on a Hilbert space H. Prove that for a densely defined closed operator T, the spectrum (T) is closed and non-empty. Discuss the challenges of extending spectral theory to unbounded operators, and prove that if I is self- adjoint, then (T) CR

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://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 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. This is same as writing f(x) de 2. Can you guess what the symbol 2 f(x) means? LIIS RIIS 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 Stone-Weierstrass Theorem in Spectral Theory Let C(X) denote the space of continuous complex-valued functions on a compact Hausdorff space X. Prove the Stone-Weierstrass theorem, which states that any subalgebra of C(X) that separates points, contains the constant functions, and is closed under complex conjugation, is dense in C(X) under the sup norm. Then, discuss how this theorem can be applied in spectral theory to approximate operators. Unbounded Operators and the Spectrum Define the spectrum of an unbounded operator T on a Hilbert space H. Prove that for a densely defined closed operator T, the spectrum (T) is closed and non-empty. Discuss the challenges of extending spectral theory to unbounded operators, and prove that if I is self- adjoint, then (T) CR

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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