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. Burkinslaw, 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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 KKR 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 z in the domain. 'This is same as writing f(x) def. Can you guess what the symbol r²: 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. 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 Let {T} be a sequence of bounded linear operators on a Hilbert space H. 1. Prove that if {T} is weakly convergent for each z EH, then {T} is uniformly bounded (use the Uniform Boundedness Principle). 2. Show that the pointwise limit T(z) = lim Ta (if it exists) defines a bounded linear operator T on H. 3. Prove that the strong convergence of {T} to T implies weak convergence, but the converse does not hold Hint: Use Banach-Steinhaus theorem, weak compactness, and examples from weak and strong operator topology.