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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 IK = R or K = C. We use the symbol, for example, f(x)=r to say that the function f is defined by setting f(x)=2 for all ir in the domain. This is same as writing f(x) de 2. Can you guess what the symbol 2: 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 1. Spectral Radius of Bounded Operators Let T be a bounded linear operator on a complex Banach space X. Prove that the spectral radius of T, defined by r(T) = sup{|A|: A € (T)}, satisfies r(T) = lim |T|| 8042 Provide a detailed proof using the Gelfand formula and consider whether the spectral radius can be characterized similarly in other types of spaces. 2. Compact Operators and Spectral Properties Let T be a compact operator on an infinite-dimensional Hilbert space H. Show that any non- zero element of the spectrum of T is an eigenvalue with finite multiplicity, and that 0 is the only possible accumulation point of the spectrum. In your proof, use the Fredholm alternative and illustrate how compactness of T influences the spectral properties compared to non-compact operators.