Robert Phelps

Phelps wrote his dissertation on subreflexive Banach spaces under the supervision of Victor Klee in 1958 at the University of Washington.{{MathGenealogy|id=28156}} Phelps was appointed to a position at Washington in 1962.[http://www.math.washington.edu/People/fac_individ.php?mathid=phelps University of Washington description of Phelps]

In 2012 he became a fellow of the American Mathematical Society.[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-05-05.

He was a convinced atheist.{{Cite web | url=http://experimentalmath.info/blog/2013/02/in-memoriam-robert-r-phelps/ |title = In Memoriam: Robert R. Phelps (1926-2013) « Math Drudge}}

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 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).{{harvtxt|Ekeland|1979|p=443}}

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.{{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}} 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.{{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}}{{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}}{{harvtxt|Rao|2002}} A revised and expanded version of his Lectures on Choquet theory was republished as {{harvtxt|Phelps|2002}}.

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."Page iii of the first (1989) edition of {{harvtxt|Phelps|1993}}. 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, 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).{{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}} 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.{{harvtxt|Nashed|1990}} Now, the study of differentiability is a central concern in nonlinear functional analysis.Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.{{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}}

Phelps has published articles under the pseudonym of John Rainwater.

{{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}}

• {{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

}}

{{reflist|30em}}

• {{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}}

• [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: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