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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 z in the domain. This is same as writing f(x) 2. Can you guess what the symbol a² 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 3. Hahn-Banach Extension Theorem in Complex Vector Spaces Let X be a complex Banach space, and let p: X→R be a sublinear functional. Let f : M → C be a bounded linear functional defined on a linear subspace MCX such that Re(f(x)) < p(x) for all M. Prove that there exists an extension F: XC off to all of X such that Re(F(x)) p(x) for all x € X. • Hint: Use Zorn's Lemma to extend f to a maximal subspace and then apply analytic arguments to handle the complex case. 4. Bounded Linear Functionals and Dual Spaces Let X be a Banach space, and let X* denote its dual space. Prove that if f€ X* and ƒ 0, then there exists a €X such that ||x|| = 1 and f(x) = ||||. Use this to deduce that every element of X* attains its norm on the closed unit ball of X if X is a reflexive Banach space. ⚫ Hint: Use the Hahn-Banach theorem and consider the weak*-topology on 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) = r² to say that the function f is defined by setting f(x)=r2 for all z in the domain. This is same as writing f(x) 2. Can you guess what the symbol a² 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 3. Hahn-Banach Extension Theorem in Complex Vector Spaces Let X be a complex Banach space, and let p: X→R be a sublinear functional. Let f : M → C be a bounded linear functional defined on a linear subspace MCX such that Re(f(x)) < p(x) for all M. Prove that there exists an extension F: XC off to all of X such that Re(F(x)) p(x) for all x € X. • Hint: Use Zorn's Lemma to extend f to a maximal subspace and then apply analytic arguments to handle the complex case. 4. Bounded Linear Functionals and Dual Spaces Let X be a Banach space, and let X* denote its dual space. Prove that if f€ X* and ƒ 0, then there exists a €X such that ||x|| = 1 and f(x) = ||||. Use this to deduce that every element of X* attains its norm on the closed unit ball of X if X is a reflexive Banach space. ⚫ Hint: Use the Hahn-Banach theorem and consider the weak-topology on 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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