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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Compler Analysis, TMH Edition, 1973. Throughout these notes, we let IK R 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) do 22. Can you guess what the symbol 2: f(x) means? LIIS RIIS means that RIIS 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 Problem 18: Unbounded Operators and Closed Graph Theorem Let T: D(T) CHH be a densely defined linear operator on a Hilbert space H. 1. Prove that if T is a closed operator, then 7", the adjoint of T, is also densely defined. 2. Show that if T is symmetric (ie, (T(z), y) = (x,T(y)) for all 2, y = D(T)), then Tis closable. 3. Prove that if T' is a closed, densely defined operator with TT=I (the identity), then 'T' is surjective. Hint: Use properties of densely defined operators, adjoint operators, and symmetry arguments.

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://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Real and Compler Analysis, TMH Edition, 1973. Throughout these notes, we let IK R 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) do 22. Can you guess what the symbol 2: f(x) means? LIIS RIIS means that RIIS 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 Problem 18: Unbounded Operators and Closed Graph Theorem Let T: D(T) CHH be a densely defined linear operator on a Hilbert space H. 1. Prove that if T is a closed operator, then 7", the adjoint of T, is also densely defined. 2. Show that if T is symmetric (ie, (T(z), y) = (x,T(y)) for all 2, y = D(T)), then Tis closable. 3. Prove that if T' is a closed, densely defined operator with TT=I (the identity), then 'T' is surjective. Hint: Use properties of densely defined operators, adjoint operators, and symmetry arguments.

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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