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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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, f(x) = r² to say that the function f is defined by setting f(x) = 2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 LIIS RIIS means that RIIS is defined by LIIS. f(x) means? 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. Hence 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 1. Spectral Properties of Compact Operators on Hilbert Spaces Let T: H→ H be a compact operator on a Hilbert space H. Show that the spectrum σ(T) of T is either finite or consists of a sequence of eigenvalues converging to zero. Furthermore, prove that if Aσ(T)\{0}, then A is an eigenvalue of T, and the corresponding eigenspace is finite- dimensional. ⚫ Hint: Consider the implications of the compactness of T on the sequence {T()} for a bounded sequence {n} in H, and utilize the spectral theorem for compact operators in Hilbert spaces. 2. Open Mapping Theorem and Applications Let X and Y be Banach spaces, and let T: XY be a surjective continuous linear operator. Prove the Open Mapping Theorem, which states that I maps open sets in X onto open sets in Y. After proving the theorem, apply it to show that if T is a continuous bijective linear map between Banach spaces, then T-1 is also continuous. ⚫Hint: Start by examining the image under T of the open unit ball in X and argue by contradiction.

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://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, f(x) = r² to say that the function f is defined by setting f(x) = 2 for all in the domain. This is same as writing f(x) def 2. Can you guess what the symbol 2 LIIS RIIS means that RIIS is defined by LIIS. f(x) means? 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. Hence 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 1. Spectral Properties of Compact Operators on Hilbert Spaces Let T: H→ H be a compact operator on a Hilbert space H. Show that the spectrum σ(T) of T is either finite or consists of a sequence of eigenvalues converging to zero. Furthermore, prove that if Aσ(T)\{0}, then A is an eigenvalue of T, and the corresponding eigenspace is finite- dimensional. ⚫ Hint: Consider the implications of the compactness of T on the sequence {T()} for a bounded sequence {n} in H, and utilize the spectral theorem for compact operators in Hilbert spaces. 2. Open Mapping Theorem and Applications Let X and Y be Banach spaces, and let T: XY be a surjective continuous linear operator. Prove the Open Mapping Theorem, which states that I maps open sets in X onto open sets in Y. After proving the theorem, apply it to show that if T is a continuous bijective linear map between Banach spaces, then T-1 is also continuous. ⚫Hint: Start by examining the image under T of the open unit ball in X and argue by contradiction.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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