Instructions to follow: * Give original work Chatgpt means downvote, *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 S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KC. We use the symbol, for example, f(x)² to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) 2. Can you guess what the symbol x2f(x) means? LIIS IS 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. 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 Let be ao-finite measure absolutely continuous with respect to another o-finite measure on the same measurable space (X, F). Prove the Radon-Nikodym theorem, which states that there exists measurable function f: X(0,00) such that (A) = f du for all € F. Provide a detailed proof of existence and uniqueness off, and discuss any major lemmas used (e.g the Hahn decomposition or the Radon-Nikodym Lemma). Hint: Use the concept of absolute continuity and provide an example application of the Radon- Nikodym derivative

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