Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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Instructions to follow: * Give original work Chatgpt means downvote, *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 S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KKR or KC. We use the symbol, for example f(x)=2 to say that the function f is defined by setting f(x)=2 for all in the domain This is same as writing f(x) de 2. Can you guess what the symbol a2f(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. llene. the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 21. Spectral Theory - Stone's Theorem on One-Parameter Unitary Groups Let {U(t)}ER be a strongly continuous one-parameter unitary group on a Hilbert space H. Prove Stone's Theorem, which states that there exists a unique self-adjoint operator A on H such that U(t) et for all t = R. Provide a detailed proof, including the construction of the generator A and a discussion on the domain of A. Hint: Carefully develop the connection between unitary groups and self-adjoint operators, using spectral theory. 22. Measure Theory - Riesz Representation Theorem for LP Spaces Prove the Riesz Representation Theorem for the dual of LP(X, μ) for 1 < p < ∞ on a σ-finite measure space (X, F, μ). Show that for every bounded linear functional on LP(X,μ), there exists a unique g € L'(X, μ) (where 1/p+1/q = 1) such that (f) = f(x)g(x) du(2 ) dμ(x) for all ƒ € LP(X,μ). Include all necessary steps to establish the existence and uniqueness of g and rigorously prove each part.
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Transcribed Image Text:Instructions to follow: * Give original work Chatgpt means downvote, *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://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 KKR or KC. We use the symbol, for example f(x)=2 to say that the function f is defined by setting f(x)=2 for all in the domain This is same as writing f(x) de 2. Can you guess what the symbol a2f(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. llene. the emphasis all through had been to look at concrete spaces of function and linear map between them. This has two advantages: (1) the students get to see the typical application of the results of functional analysis to other parts of analysis and (2) while dealing with suc 21. Spectral Theory - Stone's Theorem on One-Parameter Unitary Groups Let {U(t)}ER be a strongly continuous one-parameter unitary group on a Hilbert space H. Prove Stone's Theorem, which states that there exists a unique self-adjoint operator A on H such that U(t) et for all t = R. Provide a detailed proof, including the construction of the generator A and a discussion on the domain of A. Hint: Carefully develop the connection between unitary groups and self-adjoint operators, using spectral theory. 22. Measure Theory - Riesz Representation Theorem for LP Spaces Prove the Riesz Representation Theorem for the dual of LP(X, μ) for 1 < p < ∞ on a σ-finite measure space (X, F, μ). Show that for every bounded linear functional on LP(X,μ), there exists a unique g € L'(X, μ) (where 1/p+1/q = 1) such that (f) = f(x)g(x) du(2 ) dμ(x) for all ƒ € LP(X,μ). Include all necessary steps to establish the existence and uniqueness of g and rigorously prove each part.
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