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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: Weak and Strong Convergence in I? Spaces C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let {f} CLP (R) 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: Weak and Strong Convergence in I? Spaces C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let {f} CLP (R) for 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 59E
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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: Weak and Strong Convergence in I? Spaces C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let {f} CLP (R) for 1<p< ∞, and suppose fr→f weakly in LP (R). Prove that (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) |||||, ≤lim infx||fn|p (ie, weak convergence in IP does not increase the IP-norm). E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Show that if ||fn|lp||fl|p, then fr →f strongly in LP (R). Provide examples illustrating (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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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," def f(x) = r² to say that the function f is defined by setting f(x) = r² for all 2 in the domain. This is same as writing f(x) 2. Can you guess what the symbol a 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. 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 these results and discuss the cases where strong convergence fails despite weak convergence. Closed Subspace and Projection Theorem Let H be a Hilbert space, and let M C H be a closed subspace. Prove that there exists a unique bounded linear operator P: HH such that P is an orthogonal projection onto M (ie, P(z) € M and P(a) =z for all z € M, and P(z) = 0 for all z € M+). Prove that P is self-adjoint and satisfies P2 = P. Discuss the implications of this theorem for best approximations in Hilbert spaces.
Transcribed Image Text: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: Weak and Strong Convergence in I? Spaces C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let {f} CLP (R) for 1<p< ∞, and suppose fr→f weakly in LP (R). Prove that (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) |||||, ≤lim infx||fn|p (ie, weak convergence in IP does not increase the IP-norm). E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, Show that if ||fn|lp||fl|p, then fr →f strongly in LP (R). Provide examples illustrating (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," def f(x) = r² to say that the function f is defined by setting f(x) = r² for all 2 in the domain. This is same as writing f(x) 2. Can you guess what the symbol a 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. 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 these results and discuss the cases where strong convergence fails despite weak convergence. Closed Subspace and Projection Theorem Let H be a Hilbert space, and let M C H be a closed subspace. Prove that there exists a unique bounded linear operator P: HH such that P is an orthogonal projection onto M (ie, P(z) € M and P(a) =z for all z € M, and P(z) = 0 for all z € M+). Prove that P is self-adjoint and satisfies P2 = P. Discuss the implications of this theorem for best approximations in Hilbert spaces.
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