**Robert Ralph Phelps born March 22, 1926 died on January 4, 2013 aged 86**

After an earlier career as a radio operator in the merchant marines, Bob Phelps studied at the University of California in Los Angeles and then went on to completed a PhD from the University of Washington in 1958 under the supervision of Victor Klee. His thesis was entitled “Subreflexive normed linear spaces”. (A class of Banach spaces that disappeared when Bishop and Phelps showed all Banach spaces enjoyed this very important property!) After spending two years at the Institute of Advanced Study in Princeton and a further two years at the University of California in Berkeley, he obtained a position in 1962 at the University of Washington in Seattle where he continued to teach until his retirement, becoming Emeritus in 1996. The academic year of 1969-70 was spent at the Pierre-and-Marie-Curie University (Jussieu Campus, known as `Paris VI’) on the invitation of Gustave Choquet with whom he enjoyed a friendly relationship.

Bob Phelps was an outstanding analyst; leaving behind a legacy of results that continue to have a strong impact in functional analysis and the theory of Banach spaces, including optimization. Besides publishing more than 70 papers he authored three highly influential books and sets of lecture notes:

*Lectures on Choquet’s theorem,*(Springer 1966, 2nd edition 2001) which provides an easily penetrated account of the theory of integral representations of convex sets. In the most recent edition he considered applications to approximation theory; including, for instance, Korovkin’s theorem and its extension due to Shashkin. Each of the us recall studying from this book as students and one (Thera) took part in Gustave Choquet’s the course and seminar in which the analysis had been initiated.*Lectures on the Differentiability of convex functions on Banach spaces*(1977-78),*Convex functions, monotone operators and differentiability*(Springer 1989, 2nd edition 1993), has become a highly cited classic. In the second edition he integrated recent results due to D. Preiss, R. Haydon, J.M. Borwein, S. Fitzpatrick, P. Kenderov and new work of S. Simons. This book which treats, among other things, the variational principles of Ekeland, Borwein-Preiss, Deville, Godefroy and Zizler, has provided many valuable insights for our work as researchers.

Bob collaborated with many eminent mathematicians: Isaac Namioka (a long standing colleague at the University of Washington), Joram Lindenstrauss, David Preiss, Simon Fitzpatrick (who was his student), Bernardo Cascales, Jose Orihuela, Gilles Godefroy, Frank Bonsall, David Larman, Stephen Simons and Jean-Baptiste Hiriart-Urruty. With the last of these he published an article in the Journal of Functional Analysis establishing a general formula for the subdifferential of the sum of two convex semi-continuous functions without any additional conditions using only epsilon-subdifferentials. A result important to work by two of us (Borwein, Thera).

The best known result of Bob Phelps is probably the celebrated *Bishop-Phelps theorem *obtained in collaboration with Errett Bishop, a student of Paul Halmos, better known for his “constructivist” stance. * *On Bob Phelps’ website one can find an article explaining the genesis of this result which quickly became one of the corner stones of Banach space theory. The theorem establishes that the support functionals to a closed convex nonempty bounded set C in a Banach space E are dense in the cone of linear functionals bounded from below on $C$} has proved basic to many areas including the theory of optimization. And, has been the source of several celebrated generalizations, for example, the theorem of Brondsted & Rockafellar, which shows that *if f is a lower semi-continuous convex extended-real-valued function on a real Banach space E, then the set of points where the subdifferential of f is nonempty is dense in the domain of the **functional (the set of points where the function is finite).*

In 1986, Phelps travelled to Montreal to give the opening lecture in in a workshop at CRM on differentiability of Lipschitz functions organized by John Giles and one of us (Borwein). In those days Americans frequently travelled to Canada with little ID. When Bob showed his driver’s license he was sent into the office of the head of immigration at Montreal airport. The man asked what he was doing. Bob explained and showed the meeting program. The official noted Bob’s title was “On the Bishop-Phelps theorem.” He said “Oh, you have a theorem named after you, is it any good?” Bob modestly replied that it had proven useful, at which point the man stamped an entry visa on the program and said “Welcome to Canada.” Bob thanked him and asked why he had been stopped. The reply: “We meet the most interesting people this way.”

The Phelps extremization principle applied to the epigraph of a lower semi-continuous function allows us to derive an elegant proof of the Ekeland variational principle in Banach space. This intuition yields many more recent variational principles.

Bob was an active member of the Department of Mathematics at the University of Washington. He served as Department Chair for several years and together with his wife Elaine, who was a linguist, endowed the *Robert R. and Elaine F. Phelps Professorship*. Bob Phelps was a convinced atheist and, rare for an American, almost militant in his views. He was also an accomplished sportsman; a runner, marathon walker and mountaineer par excellent. He ran every day, rain-hail or shine and, as one of us (Thera) remembers from aconference organized in Bulgaria, at a pace that no untrained colleague could have maintained.

At the University of Washington Bob founded and lead a regular research seminar known as the *Rainwater Seminar* (named after Mount Rainier, a snow draped dormant volcano iconic of Washington State and visible from his office, or was it named to reflect the nature of Seattle’s inclement climate). In deference to the Australian accent, while one of us (Sims) and several other Australians were visiting in the late seventies Bob in keeping with his wry sense of humour changed the name to “Rinewater”. Many of the significant results in functional analysis from the latter part of the 20th century were first announced at these seminars and results emerging from discussions during and after the seminar were often published under the name of “John Rainwater”.

We close with an amusing account that Bob passed on to J.-B. Hiriart-Urruty.

In the sixties: he had been invited by G. Choquet toa rather posh restaurant in Paris for dinner where he was served an exquisite piece of meat, too large for him to quite finish. He considered taking it home to have later, and thought to himself that back in the US he would simply have asked for a “Doggy Bag”, but knew that in Paris taking away the remainders of a meal was a no-no, as it is even today. Never mind, he gingerly calls a waiter and tells him in a whisper that he would like to take the meat on his plate … for his dog. The waiter swiftly removes the plate saying in a confident tone: “Sir, I have something much better for your dog, I’ll bring you some bones!”.

With the passing of Bob Phelps we lose an outstanding mathematician, sportsman and a modest and deeply humanistic colleague. He will be much missed.

Michel Thera, Brailey Sims and Jon Borwein

**Post Script.** We have received various notes from former colleagues, friends and students of Bob. Stephen Simons wrote:

I have a little story that tells you what Bob did after retirement (using the skills he developed during the war as a radio operator). A few years ago, I visited him in Seattle and I noticed some old clothes and other assorted bits of junk strewed over the back seat of his car. He told me that this junk was actually covering more than $500 of sophisticated communications equipment. He explained that he had a position in the local organization that was prepared for a (terrorist, seismic or volcanic) disaster.

As you say, he will be missed.

On March 3, 2013 this obituary appeared in the Seattle Times.