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) Hahn-Banach Theorem Extensions Question: State and prove the Hahn-Banach theorem in both its algebraic and analytic forms. Then, provide a proof of the theorem when applied to complex vector spaces. Conclude by discussing an application of the Hahn-Banach theorem in dual space theory. 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, Banach-Alaoglu Theorem (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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd..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 KR or KC. We use the symbol, for example, Question: Prove the Banach-Alaoglu theorem, which states that the closed unit ball in the dual space of a normed vector space is compact in the weak-* topology. Use Tychonoff's theorem as part of your proof, and explore its applications in reflexive spaces. f(x)=r to say that the function f is defined by setting f(x)=2 for all in the domain. Uniform Boundedness Principle (Banach-Steinhaus Theorem) def T'his is same as writing f(x) 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 Question: Let {T} be a sequence of bounded linear operators from a Banach space X to a normed space Y. Prove the Uniform Boundedness Principle, and demonstrate that if sup ||T() < ∞ for each a ЄX, then sup ||T|| < ∞o. Include an example that illustrates a situation where the Uniform Boundedness Principle applies.