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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumares, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) 2. Can you guess what the symbol a2f(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 28. Riesz-Thorin Interpolation Theorem Let T' be a linear operator from LP (2) + LP (S) to L (S) + L (S2), where T is bounded from L() to L () and from LP (S) to L (S2). Prove the Riesz-Thorin interpolation theorem, which states that I is bounded from LP (2) to L (2) for 1=1+and 1+for 0≤0≤1. • Hint: Use the complex interpolation method and carefully analyze the norms of T across the interpolation spaces. 29. Unconditional basis in Danach Spaces A basis {e} of a Banach space X is called unconditional if for every a EX, the series Σanen converges unconditionally (independent of the order of terms). Prove that if X is a Hilbert space, every orthonormal basis of X is an unconditional basis. Discuss whether this property holds for LP spaces for p +2. Hint: Use properties of orthonormal sets and examine differences in IP norms for bases when P+2.

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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumares, 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) 2. Can you guess what the symbol a2f(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 28. Riesz-Thorin Interpolation Theorem Let T' be a linear operator from LP (2) + LP (S) to L (S) + L (S2), where T is bounded from L() to L () and from LP (S) to L (S2). Prove the Riesz-Thorin interpolation theorem, which states that I is bounded from LP (2) to L (2) for 1=1+and 1+for 0≤0≤1. • Hint: Use the complex interpolation method and carefully analyze the norms of T across the interpolation spaces. 29. Unconditional basis in Danach Spaces A basis {e} of a Banach space X is called unconditional if for every a EX, the series Σanen converges unconditionally (independent of the order of terms). Prove that if X is a Hilbert space, every orthonormal basis of X is an unconditional basis. Discuss whether this property holds for LP spaces for p +2. Hint: Use properties of orthonormal sets and examine differences in IP norms for bases when P+2.

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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