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 S. Kumaresan, Real Analysis An Outline, Unpublished Course Notes (available at http://mtto.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).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) 2. Can you guess what the symbol a² f(a) means? LIIS RIIS means that RIIS is defined by LIIS. dof 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 9. Sobolev Spaces and Embedding Theorems Let CR" be a bounded open subset with a smooth boundary. For 1 , then WP (2) is continuously embedded in C(S), the space of continuous functions on 2. • Hint: Use the Sobolev embedding theorem and analyze how the choice of k and p affects continuity. 10. Closed Graph Theorem and Applications Let X and Y be Banach spaces, and let T: XY be a linear operator. Prove the Closed Graph Theorem, which states that if I has a closed graph (ie., if x, x in X and T(x) → y in Y imply T(x) = y), then I is continuous. Then, apply this result to show that if I is a densely defined linear operator on L²(R) with a closed graph, then I is bounded. Hint: Argue by considering sequences in X and show that the graph being closed implies the continuity of T.

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://mtto.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) 2. Can you guess what the symbol a² f(a) means? LIIS RIIS means that RIIS is defined by LIIS. dof 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 9. Sobolev Spaces and Embedding Theorems Let CR" be a bounded open subset with a smooth boundary. For 1 , then WP (2) is continuously embedded in C(S), the space of continuous functions on 2. • Hint: Use the Sobolev embedding theorem and analyze how the choice of k and p affects continuity. 10. Closed Graph Theorem and Applications Let X and Y be Banach spaces, and let T: XY be a linear operator. Prove the Closed Graph Theorem, which states that if I has a closed graph (ie., if x, x in X and T(x) → y in Y imply T(x) = y), then I is continuous. Then, apply this result to show that if I is a densely defined linear operator on L²(R) with a closed graph, then I is bounded. Hint: Argue by considering sequences in X and show that the graph being closed implies the continuity of T.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 5E

See similar textbooks