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 S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 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. 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 Dual Space and Weak Convergence of Operators Let X be a Banach space, and let T,, : XX be a sequence of bounded linear operators that converge weakly to an operator T, ie, T.(a)T(z) for all z X. Prove that if {T} is uniformly bounded (i.e., sup, ||T|| < ∞0), then I is also a bounded linear operator. Moreover, show that if Tn converges strongly to T, then T converges strongly to T* on the dual space X*. Open Mapping Theorem and Applications to Isomorphisms Let X and Y be Banach spaces, and let T: XY be a bounded linear operator. Prove that if I is surjective, then I is an open mapping (i.e., it maps open sets to open sets). Use this result to prove that if T'is bijective, then T is an isomorphism, meaning there exists a bounded inverse T-1: Y→ X. Apply the Open Mapping Theorem to solve an example problem involving isomorphisms of Banach spaces.