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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KR or KC. We use the symbol, for example, f(x)=r2 to say that the function f is defined by setting f(x) = for all in the domain. f(x) means? This is same as writing f(x) de 2. Can you guess what the symbol 2 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 11. Operator Theory - The Spectral Theorem for Unbounded Self-Adjoint Operators Let T' be an unbounded self-adjoint operator on a Hilbert space H. Prove that T has a spectral decomposition given by a projection-valued measure E on H such that T = [AdE(X). Carefully outline the construction of E and justify each step using the theory of unbounded operators and the spectral theorem for self-adjoint operators. Hint: Discuss domains of unbounded operators, the role of dense subspaces, and regularization techniques. 12. Spectral Theory - Functional Calculus for Bounded Self-Adjoint Operators Let T be a bounded self-adjoint operator on a Hilbert space H. Construct a proof for the existence of a functional calculus that associates each continuous real-valued function f on σ(T) to an operator f(T). Show that this functional calculus is unique and satisfies f(T) is self-adjoint if f is real-valued.