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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd, (1996).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) = r² for all in the domain. This is same as writing f(x) def 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. 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 13. Functional Calculus for Bounded Normal Operators Let T be a bounded normal operator on a Hilbert space H, and let σ (T) denote its spectrum. Prove that there exists a continuous functional calculus for T, meaning that for every continuous function f defined on σ(I), there exists an operator f(T) such that f(T) is normal and satisfies o(f(T)) = f(o(T)). Additionally, prove that if f is bounded, then ||f(T) || ||fx • Hint: Use the spectral theorem for normal operators and construct f(T) via integration over the spectral measure. 14. Weak Convergence and Reflexive Banach Spaces Prove that a Banach space X is reflexive if and only if every bounded sequence in X has a weakly convergent subsequence. Use this result to show that LP (S) is reflexive for 1