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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. Kumar R (available:// Function age An Quiline, Unpulslied Course Not widow.loads) B.V. Lim Edition, New Age mternational Ltd., (1996) Budin Real and Compter Analysis, TMH Enion, 7 Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)=r 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) def 2. Can you guess what the symbol a2f(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. Hence 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 27. Compact Operators on Hilbert Spaces and Schatten Classes Let T be a compact operator on a Hilbert space H. For 1

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. Kumar R (available:// Function age An Quiline, Unpulslied Course Not widow.loads) B.V. Lim Edition, New Age mternational Ltd., (1996) Budin Real and Compter Analysis, TMH Enion, 7 Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)=r 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) def 2. Can you guess what the symbol a2f(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. Hence 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 27. Compact Operators on Hilbert Spaces and Schatten Classes Let T be a compact operator on a Hilbert space H. For 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 5E
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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. Kumar R (available:// Function age An Quiline, Unpulslied Course Not widow.loads) B.V. Lim Edition, New Age mternational Ltd., (1996) Budin Real and Compter Analysis, TMH Enion, 7 Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)=r 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) def 2. Can you guess what the symbol a2f(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. Hence 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 27. Compact Operators on Hilbert Spaces and Schatten Classes Let T be a compact operator on a Hilbert space H. For 1<p<∞, define the Schatten p-class SP(H) as the space of compact operators T on H such that -1 (T)" < ∞, where 8, (T) are the singular values of T. Prove that SP (H) is a Banach space and that TE S2(H) if and only if T is a Hilbert-Schmidt operator. ⚫Hint: Show that the Schatten norm ||T||,= (Σs, (T)P) 1/P satisfies completeness and relates to the Hilbert-Schmidt inner product. 28. Riesz-Thorin Interpolation Theorem Let T be a linear operator from L (S2)+LP (N) to L (N) + L (2), where T is bounded from LP (n) to L (2) and from LP () to L (2). Prove the Riesz-Thorin interpolation theorem, which states that I' is bounded from LP (2) to L" (2) for 1 = 1 + 2 and 1=1+for 00≤1. • Hint: Use the complex interpolation method and carefully analyze the norms of T across the interpolation spaces.
Transcribed Image Text: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. Kumar R (available:// Function age An Quiline, Unpulslied Course Not widow.loads) B.V. Lim Edition, New Age mternational Ltd., (1996) Budin Real and Compter Analysis, TMH Enion, 7 Throughout these notes, we let K = R or KC. We use the symbol, for example, f(x)=r 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) def 2. Can you guess what the symbol a2f(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. Hence 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 27. Compact Operators on Hilbert Spaces and Schatten Classes Let T be a compact operator on a Hilbert space H. For 1<p<∞, define the Schatten p-class SP(H) as the space of compact operators T on H such that -1 (T)" < ∞, where 8, (T) are the singular values of T. Prove that SP (H) is a Banach space and that TE S2(H) if and only if T is a Hilbert-Schmidt operator. ⚫Hint: Show that the Schatten norm ||T||,= (Σs, (T)P) 1/P satisfies completeness and relates to the Hilbert-Schmidt inner product. 28. Riesz-Thorin Interpolation Theorem Let T be a linear operator from L (S2)+LP (N) to L (N) + L (2), where T is bounded from LP (n) to L (2) and from LP () to L (2). Prove the Riesz-Thorin interpolation theorem, which states that I' is bounded from LP (2) to L" (2) for 1 = 1 + 2 and 1=1+for 00≤1. • Hint: Use the complex interpolation method and carefully analyze the norms of T across the interpolation spaces.
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