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Robert Phelps

Robert Ralph Phelps (March 22, 1926 – January 4, 2013) was an American mathematician who was known for his contributions to analysis, particularly to functional analysis and measure theory. He was a professor of mathematics at the University of Washington from 1962 until his death.

Robert R. Phelps
Born(1926-03-22)March 22, 1926
DiedJanuary 4, 2013(2013-01-04) (aged 86)
NationalityAmerican
Alma materUniversity of Washington
Known for
SpouseElaine Phelps[3]
Scientific career
Fields
InstitutionsUniversity of Washington
Doctoral advisorVictor L. Klee[1]

Biography edit

Phelps wrote his dissertation on subreflexive Banach spaces under the supervision of Victor Klee in 1958 at the University of Washington.[1] Phelps was appointed to a position at Washington in 1962.[4]

In 2012 he became a fellow of the American Mathematical Society.[5]

He was a convinced atheist.[6]

Research edit

With Errett Bishop, Phelps proved the Bishop–Phelps theorem, one of the most important results in functional analysis, with applications to operator theory, to harmonic analysis, to Choquet theory, and to variational analysis. In one field of its application, optimization theory, Ivar Ekeland began his survey of variational principles with this tribute:

The central result. The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps ... that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset XE is norm-dense in E*. The crux of the proof lies in introducing a certain convex cone in E, associating with it a partial ordering, and applying to the latter a transfinite induction argument (Zorn's lemma).[7]

Phelps has written several advanced monographs, which have been republished. His 1966 Lectures on Choquet theory was the first book to explain the theory of integral representations.[8] In these "instant classic" lectures, which were translated into Russian and other languages, and in his original research, Phelps helped to lead the development of Choquet theory and its applications, including probability, harmonic analysis, and approximation theory.[9][10][11] A revised and expanded version of his Lectures on Choquet theory was republished as Phelps (2002).[11]

Phelps has also contributed to nonlinear analysis, in particular writing notes and a monograph on differentiability and Banach-space theory. In its preface, Phelps advised readers of the prerequisite "background in functional analysis": "the main rule is the separation theorem (a.k.a. [also known as] the Hahn–Banach theorem): Like the standard advice given in mountaineering classes (concerning the all-important bowline for tying oneself into the end of the climbing rope), you should be able to employ it using only one hand while standing blindfolded in a cold shower."[12] Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of Asplund and Rockafellar, Phelps hammered into place the pitons, linked the carabiners, and threaded the top rope by which novices have ascended from the frozen tundras of topological vector spaces to the Shangri-La of Banach space theory. His University College, London (UCL) lectures on the Differentiability of convex functions on Banach spaces (1977–1978) were "widely distributed". Some of Phelps's results and exposition were developed in two books,[13] Bourgin's Geometric aspects of convex sets with the Radon-Nikodým property (1983) and Giles's Convex analysis with application in the differentiation of convex functions (1982).[10][14] Phelps avoided repeating the results previously reported in Bourgin and Giles when he published his own Convex functions, monotone operators and differentiability (1989), which reported new results and streamlined proofs of earlier results.[13] Now, the study of differentiability is a central concern in nonlinear functional analysis.[15][16] Phelps has published articles under the pseudonym of John Rainwater.[17]

Selected publications edit

  • Bishop, Errett; Phelps, R. R. (1961). "A proof that every Banach space is subreflexive". Bulletin of the American Mathematical Society. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4. MR 0123174.
  • Phelps, Robert R. (1993) [1989]. Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics. Vol. 1364 (2nd ed.). Berlin: Springer-Verlag. pp. xii+117. ISBN 3-540-56715-1. MR 1238715.
  • Phelps, Robert R. (2001). Phelps, Robert R (ed.). Lectures on Choquet's theorem. Lecture Notes in Mathematics. Vol. 1757 (Second edition of 1966 ed.). Berlin: Springer-Verlag. pp. viii+124. doi:10.1007/b76887. ISBN 3-540-41834-2. MR 1835574.
  • Namioka, I.; Phelps, R. R. (1975). "Banach spaces which are Asplund spaces". Duke Math. J. 42 (4): 735–750. doi:10.1215/s0012-7094-75-04261-1. hdl:10338.dmlcz/127336. ISSN 0012-7094.

Notes edit

  1. ^ a b Robert Phelps at the Mathematics Genealogy Project
  2. ^ Robert R. "Bob" Phelps Obituary
  3. ^ Page 21: Gritzmann, Peter; Sturmfels, Bernd (April 2008). "Victor L. Klee 1925–2007" (PDF). Notices of the American Mathematical Society. 55 (4). Providence, RI: American Mathematical Society: 467–473. ISSN 0002-9920.
  4. ^ University of Washington description of Phelps
  5. ^ List of Fellows of the American Mathematical Society, retrieved 2013-05-05.
  6. ^ "In Memoriam: Robert R. Phelps (1926-2013) « Math Drudge".
  7. ^ Ekeland (1979, p. 443)
  8. ^ Lacey, H. E. "Review of Gustave Choquet's (1969) Lectures on analysis, Volume III: Infinite dimensional measures and problem solutions". Mathematical Reviews. MR 0250013.
  9. ^ Asimow, L.; Ellis, A. J. (1980). Convexity theory and its applications in functional analysis. London Mathematical Society Monographs. Vol. 16. London-New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+266. ISBN 0-12-065340-0. MR 0623459.
  10. ^ a b Bourgin, Richard D. (1983). Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics. Vol. 993. Berlin: Springer-Verlag. pp. xii+474. doi:10.1007/BFb0069321. ISBN 3-540-12296-6. MR 0704815.
  11. ^ a b Rao (2002)
  12. ^ Page iii of the first (1989) edition of Phelps (1993).
  13. ^ a b Nashed (1990)
  14. ^ Giles, John R. (1982). Convex analysis with application in the differentiation of convex functions. Research Notes in Mathematics. Vol. 58. Boston, Mass.-London: Pitman (Advanced Publishing Program). pp. x+278. ISBN 0-273-08537-9. MR 0650456.
  15. ^ Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
  16. ^ Mordukhovich, Boris S. (2006). Variational analysis and generalized differentiation I and II. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. MR 2191745.
  17. ^ Phelps, Robert R. (2002). Melvin Henriksen (ed.). "Biography of John Rainwater". Topological Commentary. 7 (2). arXiv:math/0312462. Bibcode:2003math.....12462P.

References edit

  • Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.
  • Nashed, M. Z. (1990). "Review of 1989 first edition of Phelps's Convex functions, monotone operators and differentiability". Mathematical Reviews. Lecture Notes in Mathematics. 1364. doi:10.1007/BFb0089089. ISBN 978-3-540-50735-2. MR 0984602. Review of first edition of Phelps (1993).
  • Rao, T. S. S. R. K. (2002). Phelps, Robert R (ed.). "Review of Phelps (2002)". Mathematical Reviews. Lecture Notes in Mathematics. 1757. doi:10.1007/b76887. ISBN 978-3-540-41834-4. MR 1835574. Review of Phelps (2001).

External resources edit

robert, phelps, other, people, named, disambiguation, robert, ralph, phelps, march, 1926, january, 2013, american, mathematician, known, contributions, analysis, particularly, functional, analysis, measure, theory, professor, mathematics, university, washingto. For other people named Robert Phelps see Robert Phelps disambiguation Robert Ralph Phelps March 22 1926 January 4 2013 was an American mathematician who was known for his contributions to analysis particularly to functional analysis and measure theory He was a professor of mathematics at the University of Washington from 1962 until his death Robert R PhelpsBorn 1926 03 22 March 22 1926CaliforniaDiedJanuary 4 2013 2013 01 04 aged 86 Washington state 2 NationalityAmericanAlma materUniversity of WashingtonKnown forBishop Phelps theorem Banach spaces amp differentiability Choquet theorySpouseElaine Phelps 3 Scientific careerFieldsFunctional analysis measure theoryInstitutionsUniversity of WashingtonDoctoral advisorVictor L Klee 1 Contents 1 Biography 2 Research 3 Selected publications 4 Notes 5 References 6 External resourcesBiography editPhelps wrote his dissertation on subreflexive Banach spaces under the supervision of Victor Klee in 1958 at the University of Washington 1 Phelps was appointed to a position at Washington in 1962 4 In 2012 he became a fellow of the American Mathematical Society 5 He was a convinced atheist 6 Research editWith Errett Bishop Phelps proved the Bishop Phelps theorem one of the most important results in functional analysis with applications to operator theory to harmonic analysis to Choquet theory and to variational analysis In one field of its application optimization theory Ivar Ekeland began his survey of variational principles with this tribute The central result The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset X E is norm dense in E The crux of the proof lies in introducing a certain convex cone in E associating with it a partial ordering and applying to the latter a transfinite induction argument Zorn s lemma 7 Phelps has written several advanced monographs which have been republished His 1966 Lectures on Choquet theory was the first book to explain the theory of integral representations 8 In these instant classic lectures which were translated into Russian and other languages and in his original research Phelps helped to lead the development of Choquet theory and its applications including probability harmonic analysis and approximation theory 9 10 11 A revised and expanded version of his Lectures on Choquet theory was republished as Phelps 2002 11 Phelps has also contributed to nonlinear analysis in particular writing notes and a monograph on differentiability and Banach space theory In its preface Phelps advised readers of the prerequisite background in functional analysis the main rule is the separation theorem a k a also known as the Hahn Banach theorem Like the standard advice given in mountaineering classes concerning the all important bowline for tying oneself into the end of the climbing rope you should be able to employ it using only one hand while standing blindfolded in a cold shower 12 Phelps has been an avid rock climber and mountaineer Following the trailblazing research of Asplund and Rockafellar Phelps hammered into place the pitons linked the carabiners and threaded the top rope by which novices have ascended from the frozen tundras of topological vector spaces to the Shangri La of Banach space theory His University College London UCL lectures on the Differentiability of convex functions on Banach spaces 1977 1978 were widely distributed Some of Phelps s results and exposition were developed in two books 13 Bourgin s Geometric aspects of convex sets with the Radon Nikodym property 1983 and Giles s Convex analysis with application in the differentiation of convex functions 1982 10 14 Phelps avoided repeating the results previously reported in Bourgin and Giles when he published his own Convex functions monotone operators and differentiability 1989 which reported new results and streamlined proofs of earlier results 13 Now the study of differentiability is a central concern in nonlinear functional analysis 15 16 Phelps has published articles under the pseudonym of John Rainwater 17 Selected publications editBishop Errett Phelps R R 1961 A proof that every Banach space is subreflexive Bulletin of the American Mathematical Society 67 97 98 doi 10 1090 s0002 9904 1961 10514 4 MR 0123174 Phelps Robert R 1993 1989 Convex functions monotone operators and differentiability Lecture Notes in Mathematics Vol 1364 2nd ed Berlin Springer Verlag pp xii 117 ISBN 3 540 56715 1 MR 1238715 Phelps Robert R 2001 Phelps Robert R ed Lectures on Choquet s theorem Lecture Notes in Mathematics Vol 1757 Second edition of 1966 ed Berlin Springer Verlag pp viii 124 doi 10 1007 b76887 ISBN 3 540 41834 2 MR 1835574 Namioka I Phelps R R 1975 Banach spaces which are Asplund spaces Duke Math J 42 4 735 750 doi 10 1215 s0012 7094 75 04261 1 hdl 10338 dmlcz 127336 ISSN 0012 7094 Notes edit a b Robert Phelps at the Mathematics Genealogy Project Robert R Bob Phelps Obituary Page 21 Gritzmann Peter Sturmfels Bernd April 2008 Victor L Klee 1925 2007 PDF Notices of the American Mathematical Society 55 4 Providence RI American Mathematical Society 467 473 ISSN 0002 9920 University of Washington description of Phelps List of Fellows of the American Mathematical Society retrieved 2013 05 05 In Memoriam Robert R Phelps 1926 2013 Math Drudge Ekeland 1979 p 443 Lacey H E Review of Gustave Choquet s 1969 Lectures on analysis Volume III Infinite dimensional measures and problem solutions Mathematical Reviews MR 0250013 Asimow L Ellis A J 1980 Convexity theory and its applications in functional analysis London Mathematical Society Monographs Vol 16 London New York Academic Press Inc Harcourt Brace Jovanovich Publishers pp x 266 ISBN 0 12 065340 0 MR 0623459 a b Bourgin Richard D 1983 Geometric aspects of convex sets with the Radon Nikodym property Lecture Notes in Mathematics Vol 993 Berlin Springer Verlag pp xii 474 doi 10 1007 BFb0069321 ISBN 3 540 12296 6 MR 0704815 a b Rao 2002 Page iii of the first 1989 edition of Phelps 1993 a b Nashed 1990 Giles John R 1982 Convex analysis with application in the differentiation of convex functions Research Notes in Mathematics Vol 58 Boston Mass London Pitman Advanced Publishing Program pp x 278 ISBN 0 273 08537 9 MR 0650456 Lindenstrauss Joram and Benyamini Yoav Geometric nonlinear functional analysis Colloquium publications 48 American Mathematical Society Mordukhovich Boris S 2006 Variational analysis and generalized differentiation I and II Grundlehren Series Fundamental Principles of Mathematical Sciences Vol 331 Springer MR 2191745 Phelps Robert R 2002 Melvin Henriksen ed Biography of John Rainwater Topological Commentary 7 2 arXiv math 0312462 Bibcode 2003math 12462P References editEkeland Ivar 1979 Nonconvex minimization problems Bulletin of the American Mathematical Society New Series 1 3 443 474 doi 10 1090 S0273 0979 1979 14595 6 MR 0526967 Nashed M Z 1990 Review of 1989 first edition of Phelps s Convex functions monotone operators and differentiability Mathematical Reviews Lecture Notes in Mathematics 1364 doi 10 1007 BFb0089089 ISBN 978 3 540 50735 2 MR 0984602 Review of first edition of Phelps 1993 Rao T S S R K 2002 Phelps Robert R ed Review of Phelps 2002 Mathematical Reviews Lecture Notes in Mathematics 1757 doi 10 1007 b76887 ISBN 978 3 540 41834 4 MR 1835574 Review of Phelps 2001 External resources editProfessor Phelp s homepage at the University of Washington Robert Phelps University of Washington Archived from the original on March 16 2012 Robert Phelps at the Mathematics Genealogy Project Retrieved from https en wikipedia org w index php title Robert Phelps amp oldid 1221127133, wikipedia, wiki, book, books, library,

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