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: CD. Aliprantis and O. Burkinshaw, Principles of feel Analysis. Sed Edition, Harcourt Asia, (2000) J. Bak and D.J. Newman, Complex Avis, 2nd Edition, Springer Indian Reprint. (2009) Bartle and Sherber.. burodectory Real Analysis, 3rd edition. Wiley International (3001) E. Kreyzig. Introductory Functional Analysis with Applications, Wiley Singapore Edition, (3001). S. Kumares, Topology of Meiric Syuces, Narea. (2005). S. Kumaresan, Real Analysis - An Oustine, Unpublished Course Notes (available at http://atte.org.in/downloads) B.V. Limaye, Anctional Analysis, 2nd Edition, New Age International Ltd.. (1995). W. Budin. Real and Complex Analysis, TMII Edition, 1973. Troughout these notes, we let K-Ro KC. We use the gymbol, for example, fla) to say that the function f is defined by setting fix) or all in the domain. This is same as writing f(x). Can you guess what the symbol ? LUS US means that RUS is defined by LHS. 1(a) mess? I started with the principle that a first course in functional analysis is meant first, as a part of the general culture and scoord es ar important tool for any future analyst. Hence the emphasia all through had been in look at concrete spaces of function and linear mapa between them. This hea two audvansgea: (1) the students get to see the typical spplications 13. Functional Calculus for Bounded Normal Operators Let I be a bounded normal operator on a Hilbert space H. and let (T) denote its spectrum. Prove that there exists a continuous functional calculus for T, meaning that for every continuous function f defined on (T), there exists an occrator f(2) such that f(I) is normal and satisfics (f(T)) = f((T)). Additionally, prove that if f is bounded, then f(T)||< fL ⚫ Hint: Use the spectral theorem for normal operators and construct (T) via integration over the spectral measure. 14. Weak Convergence and Reflexive Banach Spaces Prove that a lanach space X is reflexive if and only if every bounded sequence in X has a weakly convergent subsequence. Use this result to show that L(S) is reflexive for 1

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: CD. Aliprantis and O. Burkinshaw, Principles of feel Analysis. Sed Edition, Harcourt Asia, (2000) J. Bak and D.J. Newman, Complex Avis, 2nd Edition, Springer Indian Reprint. (2009) Bartle and Sherber.. burodectory Real Analysis, 3rd edition. Wiley International (3001) E. Kreyzig. Introductory Functional Analysis with Applications, Wiley Singapore Edition, (3001). S. Kumares, Topology of Meiric Syuces, Narea. (2005). S. Kumaresan, Real Analysis - An Oustine, Unpublished Course Notes (available at http://atte.org.in/downloads) B.V. Limaye, Anctional Analysis, 2nd Edition, New Age International Ltd.. (1995). W. Budin. Real and Complex Analysis, TMII Edition, 1973. Troughout these notes, we let K-Ro KC. We use the gymbol, for example, fla) to say that the function f is defined by setting fix) or all in the domain. This is same as writing f(x). Can you guess what the symbol ? LUS US means that RUS is defined by LHS. 1(a) mess? I started with the principle that a first course in functional analysis is meant first, as a part of the general culture and scoord es ar important tool for any future analyst. Hence the emphasia all through had been in look at concrete spaces of function and linear mapa between them. This hea two audvansgea: (1) the students get to see the typical spplications 13. Functional Calculus for Bounded Normal Operators Let I be a bounded normal operator on a Hilbert space H. and let (T) denote its spectrum. Prove that there exists a continuous functional calculus for T, meaning that for every continuous function f defined on (T), there exists an occrator f(2) such that f(I) is normal and satisfics (f(T)) = f((T)). Additionally, prove that if f is bounded, then f(T)||< fL ⚫ Hint: Use the spectral theorem for normal operators and construct (T) via integration over the spectral measure. 14. Weak Convergence and Reflexive Banach Spaces Prove that a lanach space X is reflexive if and only if every bounded sequence in X has a weakly convergent subsequence. Use this result to show that L(S) is reflexive for 1

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015

1st Edition

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:HOUGHTON MIFFLIN HARCOURT

Chapter4: Writing Linear Equations

Section: Chapter Questions

Problem 14CR

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