Robert Ralph Phelps (March 22, 1926 – January 4, 2013) was an American mathematician who was known for his contributions toanalysis, particularly tofunctional analysis andmeasure 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 California
Died	January 4, 2013(2013-01-04) (aged 86) Washington state[2]
Nationality	American
Alma mater	University of Washington
Known for	Bishop–Phelps theorem Banach spaces &differentiability Choquet theory
Spouse	Elaine Phelps[3]
Scientific career
Fields	Functional analysis measure theory
Institutions	University of Washington
Doctoral advisor	Victor L. Klee[1]

Phelps wrote his dissertation onsubreflexiveBanach spaces under the supervision ofVictor 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 theAmerican Mathematical Society.^[5]

He was a convinced atheist.^[6]

WithErrett Bishop, Phelps proved theBishop–Phelps theorem, one of the most important results in functional analysis, with applications tooperator theory, toharmonic analysis, toChoquet theory, and tovariational analysis. In one field of its application,optimization theory,Ivar Ekeland began his survey ofvariational 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 spaceE which attain their maximum on a prescribed closed convex bounded subsetX⊂E is norm-dense inE*. The crux of the proof lies in introducing a certain convex cone inE, 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 1966Lectures on Choquet theory was the first book to explainthe 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 hisLectures on Choquet theory was republished asPhelps (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 ofAsplund andRockafellar, Phelps hammered into place thepitons, linked thecarabiners, and threaded thetop rope by which novices haveascended from the frozen tundras oftopological vector spaces to theShangri-La ofBanach space theory. HisUniversity College, London (UCL) lectures on theDifferentiability 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'sGeometric aspects of convex sets with the Radon-Nikodým property (1983) and Giles'sConvex 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 ownConvex 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 ofJohn Rainwater.^[17]

• 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.

• 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'sConvex functions, monotone operators and differentiability".Mathematical Reviews.MR 0984602.
• Rao, T. S. S. R. K. (2002). "Review of Phelps (2002)".Mathematical Reviews.MR 1835574.

• Professor Phelp's homepage at the University of Washington
• "Robert Phelps". University of Washington. Archived fromthe original on March 16, 2012.
• Robert Phelps at theMathematics Genealogy Project