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 S. Kumaresan, Real Analysis - An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) = x² 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 r2 LIIS RIIS means that RIIS is defined by LIIS. def f(x) means? 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 7. Gelfand Transform and Spectral Radius Let A be a commutative Banach algebra with unity. Define the Gelfand transform a of a Є A and prove that ||a||,=lim, ||a||1/ where ||a||, is the spectral radius of a. Show that the spectral radius satisfies ||a||||all and that equality holds if A is a C*-algebra. ⚫ Hint: Use properties of the spectrum and Gelfand's theory of commutative Banach algebras. 8. Uniform Boundedness Principle and Applications Let X be a Banach space, and let {T} be a sequence of bounded linear operators from X to a normed space Y. Suppose that for each a €X, the sequence {T()} is bounded in Y. Use the Uniform Boundedness Principle to prove that sup,, ||T|| <∞. Then, apply this result to show that if {T} converges pointwise to an operator T: X →Y, then T is also bounded. ⚫ Hint: Start with a contradiction by assuming sup,, ||T|| = ∞ and construct an unbounded set in X.

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) = x² 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 r2 LIIS RIIS means that RIIS is defined by LIIS. def f(x) means? 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 7. Gelfand Transform and Spectral Radius Let A be a commutative Banach algebra with unity. Define the Gelfand transform a of a Є A and prove that ||a||,=lim, ||a||1/ where ||a||, is the spectral radius of a. Show that the spectral radius satisfies ||a||||all and that equality holds if A is a C*-algebra. ⚫ Hint: Use properties of the spectrum and Gelfand's theory of commutative Banach algebras. 8. Uniform Boundedness Principle and Applications Let X be a Banach space, and let {T} be a sequence of bounded linear operators from X to a normed space Y. Suppose that for each a €X, the sequence {T()} is bounded in Y. Use the Uniform Boundedness Principle to prove that sup,, ||T|| <∞. Then, apply this result to show that if {T} converges pointwise to an operator T: X →Y, then T is also bounded. ⚫ Hint: Start with a contradiction by assuming sup,, ||T|| = ∞ and construct an unbounded set in X.

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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