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→Research: A revised and expanded version of his ''Lectures on Choquet theory'' was republished as {{harvtxt|Phelps|2002}}.
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Phelps has written several advanced monographs, which have been republished. His 1966 ''Lectures on Choquet theory'' was the first book to explain [[Choquet theory|the theory of integral representations]].<ref>{{cite journal|journal=Mathematical Reviews|last=Lacey|first=H. E.|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="Rao">{{harvtxt|Rao|2002}}</ref><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|
Phelps has written several advanced monographs, which have been republished. His 1966 ''Lectures on Choquet theory'' was the first book to explain [[Choquet theory|the theory of integral representations]].<ref>{{cite journal|journal=Mathematical Reviews|last=Lacey|first=H. E.|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="Rao">{{harvtxt|Rao|2002}}</ref><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|ref=harv}} </ref>
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|ref=harv}} </ref> 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."<ref> Page iii of the first (1989) edition of {{harvtxt|Phelps|1991}}.</ref> Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of [[Edgar Asplund|Asplund]] and [[R. Tyrrell Rockafellar|Rockafellar]], Phelps hammered into place the [[carabiner]]s and threaded the [[top roping|top rope]] by which novices have [[top roping|ascended]] from the frozen tundras of [[locally convex topological vector space|topological vector space]]s 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=harv}}</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>[[Joram Lindenstrauss|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society. </ref><ref>{{cite book|authorlink=Boris Mordukhovich|first=Boris S.|last=Mordukhovich|title=Variational analysis and generalized differentiation&nbsp;'''I''' and '''II'''|series=Grundlehren Series (Fundamental Principles of Mathematical Sciences)|volume=331|publisher=Springer|year=2006|chapter=|pages=|MR=2191745|ref=harv}}</ref>
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|1991}}.</ref> Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of [[Edgar Asplund|Asplund]] and [[R. Tyrrell Rockafellar|Rockafellar]], Phelps hammered into place the [[carabiner]]s and threaded the [[top roping|top rope]] by which novices have [[top roping|ascended]] from the frozen tundras of [[locally convex topological vector space|topological vector space]]s 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=harv}}</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>[[Joram Lindenstrauss|Lindenstrauss, Joram]] and Benyamini, Yoav. ''Geometric nonlinear functional analysis'' Colloquium publications, 48. American Mathematical Society. </ref><ref>{{cite book|authorlink=Boris Mordukhovich|first=Boris S.|last=Mordukhovich|title=Variational analysis and generalized differentiation&nbsp;'''I''' and '''II'''|series=Grundlehren Series (Fundamental Principles of Mathematical Sciences)|volume=331|publisher=Springer|year=2006|chapter=|pages=|MR=2191745|ref=harv}}</ref>

Version vom 11. April 2011, 10:26 Uhr

Vorlage:Infobox scientist Robert Ralph Phelps (born March 22, 1926) 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.

Biography

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.[2]

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 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).[3]

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.[4] 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.[5][6][7] A revised and expanded version of his Lectures on Choquet theory was republished as Vorlage:Harvtxt.

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."[8] Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of Asplund and Rockafellar, Phelps hammered into place 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,[9] 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).[7][10] 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.[9] Now, the study of differentiability is a central concern in nonlinear functional analysis.[11][12] Phelps has published articles under the pseudonym of John Rainwater.[13]

Selected publications

  • Errett Bishop, R. R. Phelps: A proof that every Banach space is subreflexive. In: Bulletin of the American Mathematical Society. 67. Jahrgang, 1961, S. 97–98.
  • Robert R. Phelps: Convex functions, monotone operators and differentiability (= Lecture Notes in Mathematics. Band 1364). Second edition of 1989 (Vorlage:MR) Auflage. Springer-Verlag, Berlin 1993, ISBN 3-540-56715-1, S. xii+117.
  • Robert R. Phelps: Lectures on Choquet's theorem (= Lecture Notes in Mathematics. Band 1757). Second edition of 1966 Auflage. Springer-Verlag, Berlin 2001, ISBN 3-540-41834-2, S. viii+124.
  • I. Namioka: Banach spaces which are Asplund spaces. In: Duke Math. J. 42. Jahrgang, Nr. 4, 1975, ISSN 0012-7094, S. 735–750.

See also

Notes

  1. ↑ Vorlage:MathGenealogy
  2. ↑ University of Washington description of Phelps
  3. ↑ Vorlage:Harvtxt
  4. ↑ H. E. Lacey: Review of Gustave Choquet's (1969) Lectures on analysis, Volume III: Infinite dimensional measures and problem solutions. In: Mathematical Reviews.
  5. ↑ Vorlage:Harvtxt
  6. ↑ L. Asimow, A. J. Ellis: Convexity theory and its applications in functional analysis (= London Mathematical Society Monographs. Band 16). Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York 1980, ISBN 0-12-065340-0, S. x+266.
  7. ↑ a b Richard D. Bourgin: Geometric aspects of convex sets with the Radon-Nikodým property (= Lecture Notes in Mathematics. Band 993). Springer-Verlag, Berlin 1983, ISBN 3-540-12296-6, S. xii+474.
  8. ↑ Page iii of the first (1989) edition of Vorlage:Harvtxt.
  9. ↑ a b Vorlage:Harvtxt
  10. ↑ John R. Giles: Convex analysis with application in the differentiation of convex functions (= Research Notes in Mathematics. Band 58). Pitman (Advanced Publishing Program), Boston, Mass.-London 1982, ISBN 0-273-08537-9, S. x+278.
  11. ↑ Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
  12. ↑ Boris S. Mordukhovich: Variational analysis and generalized differentiation&nbsp;'''I''' and '''II''' (= Grundlehren Series (Fundamental Principles of Mathematical Sciences). Band 331). Springer, 2006.
  13. ↑ Robert R. Phelps: Biography of John Rainwater. In: Melvin Henriksen (Hrsg.): Topological Commentary. 7. Jahrgang, Nr. 2, 2002 (yorku.ca).

References

External resources

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