Some Extension Theorems for Continuous Functions
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Nonlinear Spectral Theory Henning Wunderlich
Nonlinear Spectral Theory Henning Wunderlich Dr. Henning Wunderlich, Frankfurt, Germany E-mail address: [email protected] 1991 Mathematics Subject Classification. 46-XX,47-XX I would like to sincerely thank Prof. Dr. Delio Mugnolo for his valuable advice and support during the preparation of this thesis. Contents Introduction 5 Chapter 1. Spaces 7 1. Topological Spaces 7 2. Uniform Spaces 11 3. Metric Spaces 14 4. Vector Spaces 15 5. Topological Vector Spaces 18 6. Locally-Convex Spaces 21 7. Bornological Spaces 23 8. Barreled Spaces 23 9. Metric Vector Spaces 29 10. Normed Vector Spaces 30 11. Inner-Product Vector Spaces 31 12. Examples 31 Chapter 2. Fixed Points 39 1. Schauder-Tychonoff 39 2. Monotonic Operators 43 3. Dugundji and Quasi-Extensions 51 4. Measures of Noncompactness 53 5. Michael Selection 58 Chapter 3. Existing Spectra 59 1. Linear Spectrum and Properties 59 2. Spectra Under Consideration 63 3. Restriction to Linear Operators 76 4. Nonemptyness 77 5. Closedness 78 6. Boundedness 82 7. Upper Semicontinuity 84 Chapter 4. Applications 87 1. Nemyckii Operator 87 2. p-Laplace Operator 88 3. Navier-Stokes Equations 92 Bibliography 97 Index 101 3 Introduction The term Spectral Theory was coined by David Hilbert in his studies of qua- dratic forms in infinitely-many variables. This theory evolved into a beautiful blend of Linear Algebra and Analysis, with striking applications in different fields of sci- ence. For example, the formulation of the calculus of Quantum Mechanics (e.g., POVM) would not have been possible without such a (Linear) Spectral Theory at hand. -
Report for the Academic Year 1999-2000
Institute for ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1999 - 2000 PRINCETON NEW JERSEY Institute for ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1999 - 2000 EINSTEIN DRIVE PRINCETON • NEW JERSEY 08540-0631 609-734-8000 609-924-8399 (Fax) www.ias.edu Extract from the letter addressed by the Institute's Founders, Louis Bamberger and Mrs. Felix Fuld, to the Board of Trustees, dated June 4, 1930. Newark, New Jersey. h is fundamental in our purpose, and our express desire, that in the appointments to the staff and faculty, as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. 4 • BACKGROUND AND PURPOSE 7 • FOUNDERS, TRUSTEES, AND OFFICERS OF THE BOARD AND OF THE CORPORATION 10 ADMINISTRATION 12 • PRESENT AND PAST DIRECTORS AND FACULTY 15 • REPORT OF THE CHAIRMAN 18 • REPORT OF THE DIRECTOR - 22 • OFFICE OF THE DIRECTOR RECORD OF EVENTS 27 • ACKNOWLEDGMENTS 41 • REPORT OF THE SCHOOL OF HISTORICAL STUDIES FACULTY ACADEMIC ACTIVITIES MEMBERS, VISITORS, AND RESEARCH STAFF RECORD OF EVENTS 59 REPORT OF THE SCHOOL OF MATHEMATICS FACULTY ACADEMIC ACTIVITIES MEMBERS AND VISITORS RECORD OF EVENTS 75 • REPORT OF THE SCHOOL OF NATURAL SCIENCES -
Some Extension Theorems for Continuous Functions
Pacific Journal of Mathematics SOME EXTENSION THEOREMS FOR CONTINUOUS FUNCTIONS ERNEST A. MICHAEL Vol. 3, No. 4 June 1953 SOME EXTENSION THEOREMS FOR CONTINUOUS FUNCTIONS ERNEST MICHAEL 1. Introduction. In a recent paper, J. Dugundji proved [ 11, Th. 4.1] that every convex subset Y of a locally convex topological linear space has the following property: (1) If X is a metric space, A a closed subset of X, and / a continuous function from A into Y, then / can be extended to a continuous function from x into y. Let us call a topological space Y which has property (1) an absolute ex- tensor for metric spaces, and let absolute extensor for normal (or paracompact, etc.) spaces be defined analogously. According to Dugundji's theorem above, the supply of spaces which are absolute extensors for metric spaces is quite substantial, and it becomes reasonable to ask the following question: (2) Suppose that Y is an absolute extensor for metric spaces. Under what conditions is it also an absolute extensor for normal (or paracompact, etc.) spaces? Most of this paper (§§2-6) will be devoted to answering this question and related questions. The related questions arise in connection with the concepts of absolute retract, absolute neighborhood retract, and absolute neighborhood extensor (in §2 these are all defined and their interrelations and significance explained), and it is both convenient and natural to answer all the questions simultaneously. Assuming that the space Y of (2) is metrizable, we are able to answer these questions completely (thereby solving some heretofore unsolved problems of Arens [2, p.