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 S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 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 in the domain. 'This is same as writing f(x) def 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. 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 Banach Spaces and Compact Operators Let X be a Banach space, and let K: XX be a compact linear operator. Assume I - K is invertible, where I denotes the identity operator on X. Prove that the spectrum σ (K) of K is a countable set, possibly with 0 as its only accumulation point. Furthermore, prove that every nonzero (K) is an eigenvalue of K with finite multiplicity. Carefully analyze each step and provide a proof using the Fredholm alternative for compact operators. Spectral Theorem for Self-Adjoint Operators on Hilbert Spaces Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint linear operator. State and prove the Spectral Theorem for bounded self-adjoint operators on a separable Hilbert space in the context of functional calculus. Then, explicitly construct the spectral measure associated with T and demonstrate how it can be used to diagonalize T. Provide detailed steps in the construction and include examples for the special case where T has a purely discrete spectrum.

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 Oulline, 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 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 in the domain. 'This is same as writing f(x) def 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. 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 Banach Spaces and Compact Operators Let X be a Banach space, and let K: XX be a compact linear operator. Assume I - K is invertible, where I denotes the identity operator on X. Prove that the spectrum σ (K) of K is a countable set, possibly with 0 as its only accumulation point. Furthermore, prove that every nonzero (K) is an eigenvalue of K with finite multiplicity. Carefully analyze each step and provide a proof using the Fredholm alternative for compact operators. Spectral Theorem for Self-Adjoint Operators on Hilbert Spaces Let H be a separable Hilbert space, and let T: HH be a bounded self-adjoint linear operator. State and prove the Spectral Theorem for bounded self-adjoint operators on a separable Hilbert space in the context of functional calculus. Then, explicitly construct the spectral measure associated with T and demonstrate how it can be used to diagonalize T. Provide detailed steps in the construction and include examples for the special case where T has a purely discrete spectrum.

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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