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 *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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 K = C. We use the symbol, for example, f(x)=2 to say that the function f is defined by setting f(x) = for all z 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. 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 17. Uniformly Convex Spaces and Strong Convergence A Banach space X is uniformly convex if for every € > 0, there exists > 0 such that if || || = ||y|| = 1 and ||xy||≥e, then ||||≤1-8. Prove that every bounded sequence in a uniformly convex Banach space has a unique weak limit, provided it converges weakly. Use this property to show that every weakly convergent sequence in a uniformly convex Banach space converges strongly. ⚫ Hint: Analyze the geometric structure of the unit ball in uniformly convex spaces and apply Clarkson's inequalities if needed. 18. Spectral Properties of Unbounded Operators in Quantum Mechanics Let H = L²(R) and consider the operator T = -on H with domain D(T) = W2,2 (R), the Sobolev space of functions with square-integrable second derivatives. Prove that I is a self-adjoint operator and determine the spectrum of T. Discuss the implications of this result for the quantum harmonic oscillator. Hint: Use Fourier transform techniques to analyze the action of T on L2 (R) and relate it to the spectral properties of T.
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Transcribed Image Text: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 K = C. We use the symbol, for example, f(x)=2 to say that the function f is defined by setting f(x) = for all z 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. 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 17. Uniformly Convex Spaces and Strong Convergence A Banach space X is uniformly convex if for every € > 0, there exists > 0 such that if || || = ||y|| = 1 and ||xy||≥e, then ||||≤1-8. Prove that every bounded sequence in a uniformly convex Banach space has a unique weak limit, provided it converges weakly. Use this property to show that every weakly convergent sequence in a uniformly convex Banach space converges strongly. ⚫ Hint: Analyze the geometric structure of the unit ball in uniformly convex spaces and apply Clarkson's inequalities if needed. 18. Spectral Properties of Unbounded Operators in Quantum Mechanics Let H = L²(R) and consider the operator T = -on H with domain D(T) = W2,2 (R), the Sobolev space of functions with square-integrable second derivatives. Prove that I is a self-adjoint operator and determine the spectrum of T. Discuss the implications of this result for the quantum harmonic oscillator. Hint: Use Fourier transform techniques to analyze the action of T on L2 (R) and relate it to the spectral properties of T.
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