Instructions to follow: * Give original work Copy paste from chatgpt will get downvote *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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtta.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) = r² for all in the domain. This is same as writing f(x) df 2. Can you guess what the symbol 2: f(x) means? LIIS RIIS 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. llence 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 30. Dual Spaces of Sobolev Spaces Let W1 (2) be the Sobolev space on a bounded open subset CR". Show that the dual space (W¹ (2))* is isometrically isomorphic to a subspace of W-1.9 (2) where ½ += 1. Explain how this duality relates to boundary conditions on 2 and weak derivatives. • Hint: Construct functionals on WP (2) using weak derivatives and apply Hölder's inequality to show the boundedness of these functionals. 31. Krein-Milman Theorem and Extreme Points Let K be a compact convex subset of a locally convex topological vector space X. Prove the Krein- Milman theorem, which states that K is the closed convex hull of its extreme points. Apply this theorem to show that the closed unit ball in C([0, 1]), the space of continuous functions on [0, 1], has no extreme points. • Hint: Use Zorn's lemma to find the extreme points and apply the theorem in the context of C([0, 1]) to analyze the unit ball.