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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 = R or K = C. We use the symbol, for example, f(x) to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) def. Can you guess what the symbol 2: f(x) means? LIIS RIIS means that IIIS 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 15: Hilbert-Schmidt Operators and Trace Class Let H be a Hilbert space, and let T: HH be a Hilbert-Schmidt operator. 1. Show that T is compact. 2. Prove that the Hilbert-Schmidt operators form a two-sided ideal in the space of bounded operators on H. 3. Define the trace of a trace-class operator and show that the trace of a product of two Hilbert- Schmidt operators is independent of the order of multiplication (ie., Tr(AB) = Tr(BA) for trace-class operators A and B). Hint: Use properties of operators in Hilbert spaces, definitions of Hilbert-Schmidt norms, and the cyclicity of the trace.

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