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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). W. Rudin, Rea! and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let KKR 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) de 2. Can you guess what the symbol 2²: f(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 Problem 8: Open Mapping Theorem and Applications Let T: X→Y be a surjective bounded linear operator between Banach spaces X and Y. T 1. Prove the Open Mapping Theorem, showing that I maps open sets in X to open sets in Y. 2. Show that as a consequence, any surjective bounded linear operator between Banach spaces has a continuous inverse. 3. Use the Open Mapping Theorem to show that if a series T() converges in Y for a Σn=1 52n=1 sequence in X, then must converge in X. n=1 Hint: Consider Boire's category theorem and properties of dense subsets.