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 S. Kumaresan, Real Analysis - An Outline, Unpublished Course Notes (available at http://mtta.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KR or KC. 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) 2. Can you guess what the symbol 2 f(x) means? LIIS RIS means that IIIS is defined by LIIS. def 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 30. Dual Spaces of Sobolev Spaces Let W¹P(2) be the Sobolev space on a bounded open subset CR". Show that the dual space (WP (2))* is isometrically isomorphic to a subspace of W-14 (2) where +1. Explain ho this duality relates to boundary conditions on 2 and weak derivatives. • Hint: Construct functionals on W1P (2) using weak derivatives and apply Hölder's inequality t show the boundedness of these functionals. exo Points Let K be a compact convex subset of a locally convex topological vector space X. Prove the Krein Milman theorem, which states that K is the closed convex hull of its extreme points. Apply this theorem to show that the closed unit ball in C([0, 1]), the space of continuous functions on [0, 1], has no extreme points. • Hint: Use Zorn's lemma to find the extreme points and apply the theorem in the context of C([0, 1]) to analyze the unit ball.

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://mtta.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 KR or KC. 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) 2. Can you guess what the symbol 2 f(x) means? LIIS RIS means that IIIS is defined by LIIS. def 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 30. Dual Spaces of Sobolev Spaces Let W¹P(2) be the Sobolev space on a bounded open subset CR". Show that the dual space (WP (2))* is isometrically isomorphic to a subspace of W-14 (2) where +1. Explain ho this duality relates to boundary conditions on 2 and weak derivatives. • Hint: Construct functionals on W1P (2) using weak derivatives and apply Hölder's inequality t show the boundedness of these functionals. exo Points Let K be a compact convex subset of a locally convex topological vector space X. Prove the Krein Milman theorem, which states that K is the closed convex hull of its extreme points. Apply this theorem to show that the closed unit ball in C([0, 1]), the space of continuous functions on [0, 1], has no extreme points. • Hint: Use Zorn's lemma to find the extreme points and apply the theorem in the context of C([0, 1]) to analyze the unit ball.

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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