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: . Hahn-Banach Theorem and Applications to Duality C.D. Aliprantis and O. Burkinshaw, Principles of Real Analysis, 3rd Edition, Harcourt Asia, Let X be a normed vector space, and let ✗* be its dual space. Use the Hahn-Banach Theorem (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) to prove that every continuous linear functional f X* attains its norm on the unit ball in X if E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Singapore Edition, X is a reflexive Banach space. Additionally, explore the implications of this result for weak-star (2001). S. Kumaresan, Topology of Metric Spaces, Narosa, (2005). S. Kumaresan, Real Analysis - An Oulline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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," compactness in X and prove that the closed unit ball in X" is weak-star compact using Alaoglu's theorem. Weak and Strong Convergence in IP Spaces f(x): to say that the function f is defined by setting f(x) = r² for all in the domain. Let {f} CLP (R) for 1

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A: import sympy as sp # Define variables s, t = sp.symbols('s t') q_0, i_0 = sp.symbols('q_0 i_0') #…

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