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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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)= 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) 2. Can you guess what the symbol a² f(x) means? LIIS RIIS means that RIIS is defined by LIIS. def 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 Problem 23: Spectral Theorem for Unbounded Self-Adjoint Operators Let T be an unbounded, densely defined, self-adjoint operator on a Hilbert space H. 1. State and prove the Spectral Theorem for unbounded self-adjoint operators, showing that T can be represented in terms of a spectral measure E on the real line. 2. Show that if f is a Borel-measurable function on R, then f(T) is a well-defined operator on H. 3. Apply this theorem to analyze the spectrum of the differential operator T = -1on L²(R). Hint: Use functional calculus for self-adjoint operators and properties of Borel measures. Problem 24: Distribution Theory and Fourier Transforms of Distributions Consider the space of Schwartz functions S(R") and the space of tempered distributions S'(R"). 1. Define the Fourier transform F on S(R) and extend it to S'(R"). 2. Show that the Fourier transform is an isomorphism on S'(R"). 3. Prove that the Fourier transform of a derivative OT of a tempered distribution T is (i) F(T), where denotes the frequency variable.

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)= 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) 2. Can you guess what the symbol a² f(x) means? LIIS RIIS means that RIIS is defined by LIIS. def 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 Problem 23: Spectral Theorem for Unbounded Self-Adjoint Operators Let T be an unbounded, densely defined, self-adjoint operator on a Hilbert space H. 1. State and prove the Spectral Theorem for unbounded self-adjoint operators, showing that T can be represented in terms of a spectral measure E on the real line. 2. Show that if f is a Borel-measurable function on R, then f(T) is a well-defined operator on H. 3. Apply this theorem to analyze the spectrum of the differential operator T = -1on L²(R). Hint: Use functional calculus for self-adjoint operators and properties of Borel measures. Problem 24: Distribution Theory and Fourier Transforms of Distributions Consider the space of Schwartz functions S(R") and the space of tempered distributions S'(R"). 1. Define the Fourier transform F on S(R) and extend it to S'(R"). 2. Show that the Fourier transform is an isomorphism on S'(R"). 3. Prove that the Fourier transform of a derivative OT of a tempered distribution T is (i) F(T), where denotes the frequency variable.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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