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: Closed Subspace and Projection Theorem C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let H be a Hilbert space, and let MCH be a closed subspace. Prove that there exists a (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) unique bounded linear operator P: HH such that P is an orthogonal projection onto M E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, (i.e., P(x) = M and P(x) = x for all x € M, and P(x) = 0 for all x € M+). Prove that P (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresun, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 K = R or K = C. We use the symbol, for example, f(x)= r² to say that the function f is defined by setting f(x) = for all in the domain. This is same as writing f(x) of 22. Can you guess what the symbol 2: f(x) means? LIIS RIS means that RIIS is defined by LIIS. I started with the principle that a first course in functional analysis is meant first as a the emphasis all through had been to look at concrete spaces of function and linear maps part of the general culture and second as an important tool for any future analyst. llence 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 is self-adjoint and satisfies P2 = P. Discuss the implications of this theorem for best approximations in Hilbert spaces. Dual Space and Weak Convergence of Operators Let X be a Banach space, and let T : XX be a sequence of bounded linear operators that converge weakly to an operator T, i.e., T, (x)T(x) for all EX. Prove that if {T} is uniformly bounded (i.e., sup ||T|| < ∞), then T is also a bounded linear operator. Moreover, show that if T converges strongly to T, then T converges strongly to 7" on the dual space X*.

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, Open Mapping Theorem and Applications (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limayc, Functional Analysis, 2nd Edition, New Age International Ltd. (1996). W. Rudin, Real and Complex Analysis, TMH Edition, 1973. Throughout these notes, we let K = R or K = C. We use the symbol, for example, def Question: Prove the Open Mapping Theorem, which states that any continuous surjective linear operator between Banach spaces is an open map. Provide an example to illustrate this theorem and discuss one practical application in the theory of differential equations. f(x)=2 to say that the function f…

What is a linear subspace?

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one example of a 4-dimensional vector space that is not R 4

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