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, Dual Spaces and Weak Topologies I Let X be a reflexive Banach space with dual space X*. Show that the weak topology on X is از Hausdorff, and that the weak topology is strictly coarser than the norm topology on X (i.e., it J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) It has fewer open sets than the norm topology). Furthermore, prove that every bounded sequence (2000) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, t in X has a weakly convergent subsequence. Use tools like Alaoglu's Theorem and the Banach- (2001). S. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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. + Alaoglu theorem in your proof, and provide examples to illustrate your result. [L^p Spaces and Convergence of Fourier Series Throughout these notes, we let K = R or K = C. We use the symbol :, for example, Let f = LP ([0,2]) for 1 2, then the Fourier series of f converges to f in the LP-norm. If p < 2, discuss whether convergence still holds the emphasis all through had been to look at concrete spaces of function and linear maps and explore any counterexamples or additional conditions required for convergence. Include 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 proofs using the concepts of orthogonal projections and Parseval's identity.

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, Dual Spaces and Weak Topologies I Let X be a reflexive Banach space with dual space X*. Show that the weak topology on X is از Hausdorff, and that the weak topology is strictly coarser than the norm topology on X (i.e., it J. Bak and D.J. Newman, Complex Analysis, 2nd Edition, Springer Indian Reprint, (2009) It has fewer open sets than the norm topology). Furthermore, prove that every bounded sequence (2000) Bartle and Sherbert, Introductory Real Analysis, 3rd edition, Wiley International, (2001) E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, t in X has a weakly convergent subsequence. Use tools like Alaoglu's Theorem and the Banach- (2001). S. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, 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. + Alaoglu theorem in your proof, and provide examples to illustrate your result. [L^p Spaces and Convergence of Fourier Series Throughout these notes, we let K = R or K = C. We use the symbol :, for example, Let f = LP ([0,2]) for 1 2, then the Fourier series of f converges to f in the LP-norm. If p < 2, discuss whether convergence still holds the emphasis all through had been to look at concrete spaces of function and linear maps and explore any counterexamples or additional conditions required for convergence. Include 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 proofs using the concepts of orthogonal projections and Parseval's identity.

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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