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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Ouiline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KR or K = C. We use the symbol, for example, f(x)=r to say that the function f is defined by setting f(x) = for all z in the domain. This is same as writing f(x) de 2. Can you guess what the symbol 2 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 Problem Statement: Let H (R) and consider the singular integral operator T defined by f(y) Tf(x) = p.v. x-y where p.v. denotes the Cauchy principal value. 1. Self-Adjointness and Spectral Properties: Prove that I is a densely defined, unbounded self- adjoint operator on H and determine its spectrum. 2. Spectral Measure Construction: Construct the spectral measure E associated with T and demonstrate that I admits a spectral decomposition in terms of E. 3. Measure-Theoretic Analysis of Singular Integrals: Analyze the measure-theoretic aspects of the singular integral in the context of the spectral decomposition, proving that the integral representation of I respects the measure-theoretic properties of the singular kernel. Requirements: ⚫Handle unbounded singular integral operators within the spectral framework. Apply advanced techniques in measure theory to manage principal value integrals. Establish self-adjointness and spectral characteristics using functional analysis tools.