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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://atts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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 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 z in the domain. This is same as writing f(x) df 2. Can you guess what the symbol a² f(a) means? LIIS RIIS means that RIIS is defined by LIIS. Problem 22: Sobolev Embeddings and Rellich-Kondrachov Compactness Let CR be a bounded open set. 1. Prove the Sobolev Embedding Theorem: if t€ W(), then u L(2) for certain g depending on k, p, and 12. 2. Show that if p