{{Short description|American mathematician (1926–2013)}} {{Other people|Robert Phelps}}
{{Infobox scientist | name = Robert R. Phelps | image = Robert Phelps.jpg | image_size = 225px | alt = Phelps's head and upper torso—''Mens sana in corpore sano'' | caption = | birth_date = {{Birth date|1926|03|22}} <!--Facebook--> | birth_place = California | death_date = {{Death date and age|2013|01|04|1926|03|22}} | death_place = Washington state<ref>[http://www.legacy.com/obituaries/SeattleTimes/obituary.aspx?n=Robert-R-Phelps-Bob&pid=163441247#fbLoggedOut Robert R. "Bob" Phelps Obituary]</ref> | citizenship = | fields = {{plainlist| * Functional analysis * measure theory }} | workplaces = University of Washington | alma_mater = University of Washington | doctoral_advisor = Victor L. Klee<ref name="MGP"/> | academic_advisors = | doctoral_students = | notable_students = | known_for = {{plainlist| * Bishop–Phelps theorem * Banach spaces & differentiability * Choquet theory }} | awards = | signature = <!--(filename only)--> | signature_alt = | footnotes = | spouse = Elaine Phelps<ref name=amsnotices>Page 21: {{Cite journal | last1 = Gritzmann | first1 = Peter | last2 = Sturmfels | first2 = Bernd |author-link2=Bernd Sturmfels| title = Victor L. Klee 1925–2007 | journal = Notices of the American Mathematical Society | volume = 55 | issue = 4 | pages = 467–473 | publisher = American Mathematical Society | location = Providence, RI | date = April 2008 | url=https://www.ams.org/notices/200804/tx080400467p.pdf | issn = 0002-9920}}</ref><!-- Choquet theory and Facebook --> }}
'''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.
==Biography== Phelps wrote his dissertation on subreflexive Banach spaces under the supervision of Victor Klee in 1958 at the University of Washington.<ref name="MGP" >{{MathGenealogy|id=28156}}</ref> Phelps was appointed to a position at Washington in 1962.<ref>{{Cite web |url=http://www.math.washington.edu/People/fac_individ.php?mathid=phelps |title=University of Washington description of Phelps |access-date=2011-04-02 |archive-date=2015-06-15 |archive-url=https://web.archive.org/web/20150615124758/http://www.math.washington.edu/People/fac_individ.php?mathid=phelps |url-status=dead }}</ref>
In 2012 he became a fellow of the American Mathematical Society.<ref>[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-05-05.</ref>
He was a convinced atheist.<ref>{{Cite web | url=http://experimentalmath.info/blog/2013/02/in-memoriam-robert-r-phelps/ |title = In Memoriam: Robert R. Phelps (1926-2013) « Math Drudge}}</ref>
Phelps and his wife, Elaine Phelps, endowed a fund and professorships in the mathematics department at the University of Washington.<ref name="c402">{{cite web | last=Auteri | first=Francesco (Frank) P. | title=The Seattle Times | website=Legacy.com | date=2013-03-03 | url=https://www.legacy.com/us/obituaries/seattletimes/name/robert-phelps-obituary?id=13178663 }}</ref> ==Research== 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: <blockquote>''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).<ref>{{harvtxt|Ekeland|1979|p=443}}</ref></blockquote>
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.<ref>{{cite journal|journal=Mathematical Reviews|last=Lacey|first=H. E.|author-link=Howard Elton Lacey|title=Review of Gustave Choquet's (1969) ''Lectures on analysis'', Volume III: ''Infinite dimensional measures and problem solutions''|mr=250013}}</ref> 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.<ref name="AsimowEllis">{{cite book|last1=Asimow|first1=L.|last2=Ellis|first2=A. J.|title=Convexity theory and its applications in functional analysis|series=London Mathematical Society Monographs|volume=16|publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]|location=London-New York|year=1980|pages=x+266|isbn=0-12-065340-0|mr=623459}}</ref><ref name="Bourgin">{{cite book| last=Bourgin|first=Richard D.|title=Geometric aspects of convex sets with the Radon-Nikodým property|series= Lecture Notes in Mathematics|volume=993|publisher=Springer-Verlag|location=Berlin|year=1983|pages=xii+474|isbn=3-540-12296-6|mr=704815|doi=10.1007/BFb0069321}}</ref><ref name="Rao">{{harvtxt|Rao|2002}}</ref> A revised and expanded version of his ''Lectures on Choquet theory'' was republished as {{harvtxt|Phelps|2002}}.<ref name="Rao"/>
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."<ref>Page iii of the first (1989) edition of {{harvtxt|Phelps|1993}}.</ref> 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.<!-- The prose is purple, but consider: Phelps climbs rocks, and refers to the Hahn Banach theorem as the "bowline" of functional analysis! --> 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,<ref name="Nashed "/> 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).<ref name="Bourgin"/><ref>{{cite book|last=Giles|first=John R.|title=Convex analysis with application in the differentiation of convex functions|series=Research Notes in Mathematics|volume=58|publisher=Pitman (Advanced Publishing Program)|location=Boston, Mass.-London|year=1982|pages=x+278|isbn=0-273-08537-9|mr=650456}}</ref> 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.<ref name="Nashed">{{harvtxt|Nashed|1990}}</ref> Now, the study of differentiability is a central concern in nonlinear functional analysis.<ref>Lindenstrauss, Joram and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society.</ref><ref>{{cite book|author-link=Boris Mordukhovich|first=Boris S.|last=Mordukhovich|title=Variational analysis and generalized differentiation '''I''' and '''II'''|series=Grundlehren Series (Fundamental Principles of Mathematical Sciences)|volume=331|publisher=Springer|year=2006|mr=2191745}}</ref> Phelps has published articles under the pseudonym of John Rainwater.<ref>{{cite journal|title=Biography of John Rainwater|first=Robert R.|last=Phelps|journal=Topological Commentary|volume=7|number=2|year=2002|editor=Melvin Henriksen|url=http://at.yorku.ca/t/o/p/d/47.htm|bibcode=2003math.....12462P|arxiv=math/0312462|archive-date=2011-03-13|access-date=2011-04-02|archive-url=https://web.archive.org/web/20110313131458/http://at.yorku.ca/t/o/p/d/47.htm|url-status=dead}}</ref>
==Selected publications== * {{cite journal|last1=Bishop|first1=Errett|author-link1=Errett Bishop|last2=Phelps|first2=R. R.|title=A proof that every Banach space is subreflexive|journal=Bulletin of the American Mathematical Society|volume=67|year=1961|pages=97–98|mr=123174|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free}} * {{cite book|last=Phelps|first=Robert R.|title=Convex functions, monotone operators and differentiability|edition=2nd|orig-year= 1989 |series=Lecture Notes in Mathematics|volume=1364|publisher=Springer-Verlag|location=Berlin|year=1993|pages=xii+117|isbn=3-540-56715-1|mr=1238715}} * {{cite book|last=Phelps|first=Robert R.|editor1-first=Robert R|editor1-last=Phelps|title=Lectures on Choquet's theorem|edition=Second edition of 1966|series=Lecture Notes in Mathematics|volume=1757|publisher=Springer-Verlag|location=Berlin|year=2001|pages=viii+124|isbn=3-540-41834-2|mr=1835574|doi=10.1007/b76887}} * {{cite journal | last = Namioka | first = I.|author1-link= Isaac Namioka |author2=Phelps, R. R. | title = Banach spaces which are Asplund spaces | journal = Duke Math. J. | volume = 42 | year = 1975 | issue = 4 | pages = 735–750 | issn = 0012-7094 | doi=10.1215/s0012-7094-75-04261-1| hdl = 10338.dmlcz/127336| hdl-access = free }}
==Notes== {{reflist|30em}}
==References== * {{cite journal|last=Ekeland|first=Ivar|author-link=Ivar Ekeland|title=Nonconvex minimization problems|journal=Bulletin of the American Mathematical Society|series=New Series|volume=1|year=1979|number=3|pages=443–474|doi=10.1090/S0273-0979-1979-14595-6|mr=526967|doi-access=free}} * {{cite journal|last=Nashed|first=M. Z.|title=Review of 1989 first edition of Phelps's ''Convex functions, monotone operators and differentiability''|journal=Mathematical Reviews|mr=984602|year=1990}} * {{cite journal|first=T. S. S. R. K.|last=Rao|title=Review of Phelps (2002)|journal=Mathematical Reviews|year=2002|mr=1835574}}
==External resources== * [https://web.archive.org/web/20110303144954/http://www.math.washington.edu/~phelps/index.html Professor Phelp's homepage at the University of Washington] * {{cite web|url=http://www.math.washington.edu/People/fac_individ.php?mathid=phelps |title=Robert Phelps |publisher=University of Washington |archive-url=https://web.archive.org/web/20120316004332/http://www.math.washington.edu/People/fac_individ.php?mathid=phelps |archive-date=March 16, 2012}} * {{MathGenealogy|id=28156}}
{{Authority control}}
{{DEFAULTSORT:Phelps, Robert R.}} Category:1926 births Category:2013 deaths
Category:Sailors from California Category:Functional analysts Category:Measure theorists Category:United States Merchant Mariners Category:University of Washington faculty Category:20th-century American mathematicians Category:21st-century American mathematicians Category:Variational analysts Category:Fellows of the American Mathematical Society