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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresun, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). Question: Prove the open mapping theorem, stating that any surjective bounded linear operator between Banach spaces is an open map. Using this theorem, prove the closed graph theorem, which asserts that a linear operator between Banach spaces is continuous if its graph is closed in the product topology. Conclude with a practical application of these theorems in solving linear equations in Banach spaces. Hint: For the open mapping theorem, consider using the Baire category theorem. For the closed graph theorem, show that a closed graph implies boundedness by contradiction.
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. Kumaresun, Topology of Metric Spaces, Narosa, (2005). S. Kumaresun, Real Analysis An Outline, Unpublished Course Notes (available at http://mtts.org.in/downloads) B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996). Question: Prove the open mapping theorem, stating that any surjective bounded linear operator between Banach spaces is an open map. Using this theorem, prove the closed graph theorem, which asserts that a linear operator between Banach spaces is continuous if its graph is closed in the product topology. Conclude with a practical application of these theorems in solving linear equations in Banach spaces. Hint: For the open mapping theorem, consider using the Baire category theorem. For the closed graph theorem, show that a closed graph implies boundedness by contradiction.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.2: Representing Data
Problem 11PPS
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Transcribed Image Text: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. Kumaresun, Topology of Metric Spaces, Narosa, (2005).
S. Kumaresun, Real Analysis An Outline, Unpublished Course Notes
(available at http://mtts.org.in/downloads)
B.V. Limaye, Functional Analysis, 2nd Edition, New Age International Ltd., (1996).
Question: Prove the open mapping theorem, stating that any surjective bounded linear
operator between Banach spaces is an open map. Using this theorem, prove the closed
graph theorem, which asserts that a linear operator between Banach spaces is continuous if
its graph is closed in the product topology. Conclude with a practical application of these
theorems in solving linear equations in Banach spaces.
Hint: For the open mapping theorem, consider using the Baire category theorem. For the
closed graph theorem, show that a closed graph implies boundedness by contradiction.
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