Robert Phelps
Encyclopedia
Robert Ralph Phelps is an American mathematician who is known for his contributions to analysis
, particularly to functional analysis
and measure theory. He has been a professor of mathematics at the University of Washington since 1962.
Banach space
s under the supervision of Victor Klee
in 1958 at the University of Washington. Phelps was appointed to a position at Washington in 1962.
, 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 principle
s with this tribute:
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
. 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. A revised and expanded version of his Lectures on Choquet theory was republished as .
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." Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of Asplund and Rockafellar
, Phelps hammered into place the piton
s, linked the carabiner
s, and threaded the top rope
by which novices have ascended
from the frozen tundras of topological vector space
s 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). 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. Now, the study of differentiability is a central concern in nonlinear functional analysis.
Phelps has published articles under the pseudonym of John Rainwater
.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, particularly to functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and measure theory. He has been a professor of mathematics at the University of Washington since 1962.
Biography
Phelps wrote his dissertation on subreflexiveReflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s under the supervision of Victor Klee
Victor Klee
Victor L. Klee, Jr. was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of Washington in Seattle.Born in San Francisco, Vic Klee earned his B.A...
in 1958 at the University of Washington. Phelps was appointed to a position at Washington in 1962.
Research
With Errett BishopErrett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
, Phelps proved the Bishop–Phelps theorem, one of the most important results in functional analysis, with applications to operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, to harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, to Choquet theory
Choquet theory
In mathematics, Choquet theory is an area of functional analysis and convex analysis created by Gustave Choquet. It is concerned with measures with support on the extreme points of a convex set C...
, and to variational analysis
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. In one field of its application, optimization theory
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, Ivar Ekeland
Ivar Ekeland
Ivar Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and...
began his survey of variational principle
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
s 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).
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
Choquet theory
In mathematics, Choquet theory is an area of functional analysis and convex analysis created by Gustave Choquet. It is concerned with measures with support on the extreme points of a convex set C...
. 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. A revised and expanded version of his Lectures on Choquet theory was republished as .
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." Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of Asplund and Rockafellar
R. Tyrrell Rockafellar
* for the George Dantzig Prize in 1994 in Optima, Issue 44 page 5.- Books :* Rockafellar, R. Tyrrell. Conjugate duality and optimization. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied...
, Phelps hammered into place the piton
Piton
In climbing, a piton is a metal spike that is driven into a crack or seam in the rock with a hammer, and which acts as an anchor to protect the climber against the consequences of a fall, or to assist progress in aid climbing...
s, linked the carabiner
Carabiner
A carabiner or karabiner is a metal loop with a sprung or screwed gate that is used to quickly and reversibly connect components in safety-critical systems. The word comes from "Karabinerhaken", meaning "hook for a carbine" in German.-Use:...
s, and threaded the top rope
Top roping
Top-rope climbing is a style in climbing in which a rope, used for the climber's safety, runs from a belayer at the foot of a route through one or more carabiners connected to an anchor system at the top of the route and back down to the climber, usually attaching to the climber by means of a...
by which novices have ascended
Top roping
Top-rope climbing is a style in climbing in which a rope, used for the climber's safety, runs from a belayer at the foot of a route through one or more carabiners connected to an anchor system at the top of the route and back down to the climber, usually attaching to the climber by means of a...
from the frozen tundras of topological vector space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
s to the Shangri-La
Shangri-La
Shangri-La is a fictional place described in the 1933 novel Lost Horizon by British author James Hilton. Hilton describes Shangri-La as a mystical, harmonious valley, gently guided from a lamasery, enclosed in the western end of the Kunlun Mountains...
of Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
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). 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. Now, the study of differentiability is a central concern in nonlinear functional analysis.
Phelps has published articles under the pseudonym of John Rainwater
John Rainwater
John Rainwater is the pseudonym of a fictitious mathematician, in whose name mathematicians publish papers. Rainwater has worked mainly in functional analysis, particularly in the geometric theory of Banach spaces and in convex functions...
.