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 S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://atta.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) def. Can you guess what the symbol x2f(x) means? LIISRIIS 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 4. Measure Theory - Construction of Product Measures Let (X,A,) and (Y, B, v) be two o-finite measure spaces. Prove that there exists a unique product measure μx on the o-algebra AB generated by the measurable rectangles Ax B with A € A and B € B, such that (uxv)(Ax B) = µ(A)(B) for all A € A and B € B. Hint: Use the Carathéodory Extension Theorem, and provide detailed justifications for each step, including the uniqueness and existence of the product measure. 5. Operator Theory - Closed Operators and the Closed Graph Theorem Let T be a closed operator between two Banach spaces X and Y. Prove that if the graph of T' is closed in X x Y and T is densely defined, then I is a bounded operator. Show that this result implies the Closed Graph Theorem for operators between Banach spaces. Hint: Construct the proof by rigorously working through the definition of a closed operator, carefully showing the connection between the closed graph property and boundedness.