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) Closed Graph Theorem Proof Question: Let X and Y be Banach spaces, and let T: XY be a linear operator. Prove that if T has a closed graph, then I is continuous. Discuss the implications of this theorem for spaces that are not Banach. 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, Hahn-Banach Theorem Extensions (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) do 2. Can you guess what the symbol a²: f(x) means? LIIS RIIS means that RIIS is defined by LIIS. 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 Question: State and prove the Hahn-Banach theorem in both its algebraic and analytic forms. Then, provide a proof of the theorem when applied to complex vector spaces. Conclude by discussing an application of the Hahn-Banach theorem in dual space theory. Banach-Alaoglu Theorem Question: Prove the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed vector space is compact in the weak-* topology. Use Tychonoff's theorem as part of your proof, and explore its applications in reflexive spaces.

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) Closed Graph Theorem Proof Question: Let X and Y be Banach spaces, and let T: XY be a linear operator. Prove that if T has a closed graph, then I is continuous. Discuss the implications of this theorem for spaces that are not Banach. 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, Hahn-Banach Theorem Extensions (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtto.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) do 2. Can you guess what the symbol a²: f(x) means? LIIS RIIS means that RIIS is defined by LIIS. 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 Question: State and prove the Hahn-Banach theorem in both its algebraic and analytic forms. Then, provide a proof of the theorem when applied to complex vector spaces. Conclude by discussing an application of the Hahn-Banach theorem in dual space theory. Banach-Alaoglu Theorem Question: Prove the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed vector space is compact in the weak-* topology. Use Tychonoff's theorem as part of your proof, and explore its applications in reflexive spaces.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 14E

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