# Robert C. James
**Robert C. James** (Robert Clarke James; July 30, 1918, [Bloomington, Indiana](https://www.edgepedia.org/bloomington-indiana)) was an American mathematician whose 1950–1951 construction of a non-reflexive [Banach space](https://www.edgepedia.org/banach-space) isomorphic to its second dual, later given an equivalent norm making it isometric to its second dual, now called the James space, gave functional analysis its first example of a quasi-reflexive space.[1](https://doi.org/10.1073/pnas.37.3.174)
| Key fact | Detail |
|---|---|
| Born | July 30, 1918, Bloomington, Indiana; son of Glenn and Inez (Clarke) James[13](https://id.loc.gov/authorities/names/n84801115.html) |
| Education | B.A. UCLA 1940; Ph.D. Caltech 1947 under A. D. Michal[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506) |
| Signature result | A Banach space can be isometric with its second conjugate space without being reflexive (PNAS, communicated by J. von Neumann, December 5, 1950)[1](https://doi.org/10.1073/pnas.37.3.174) |
| The James space J | Quasi-reflexive of order one: J has codimension one in J** under the canonical injection, and J is isomorphic to J**; an equivalent norm makes it isometrically isomorphic to J**[3](https://math.colorado.edu/~rohi1040/expository/james_space.pdf) |
| What it settled | A Banach space with separable bidual need not be reflexive, and a separable space isomorphic to its bidual need not be reflexive[4](https://www.impan.pl/shop/en/publication/transaction/download/product/86355) |
| Career | Haverford 1951–1957, Harvey Mudd 1957–1967, SUNY Albany 1967–1968, Claremont Graduate School from 1968; early retirement 1981[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506) |
## Life and education
James was born in Bloomington, Indiana, to Glenn James and Inez Clarke James.[13](https://id.loc.gov/authorities/names/n84801115.html) While James was in high school, his family published the *Peace Digest*, reflecting his Quaker pacifist background.[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506) He earned his B.A. at UCLA in 1940 and his Ph.D. at Caltech in 1947 under A. D. Michal.[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506) Father and son later co-edited the *Mathematics Dictionary*, with later editions prepared with Ed Beckenbach.[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506)
## Career and academic positions
He also spent time at Harvard and UC Berkeley, and held visiting appointments at the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study) in Princeton (member 1962–1963), the Institute for Advanced Studies in Jerusalem (fellow 1976–1977), and the Mittag-Leffler Institute in Sweden.[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506) He took early retirement in 1981 and moved to [Grass Valley, California](https://www.edgepedia.org/grass-valley-california), to continue research and build a house.[2](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506)
## The James space and quasireflexive Banach spaces
In 1950 James exhibited a nonreflexive Banach space with a basis that is of finite codimension in its second dual, the first example of a quasi-reflexive space.[5](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/general-construction-of-spaces-of-the-type-of-r-c-james/C0DB90A9C18CE8B5A7B8968B37AB7339) The announcement, communicated by [John von Neumann](https://www.edgepedia.org/john-von-neumann) on December 5, 1950 and published in the *Proceedings of the National Academy of Sciences* on March 1, 1951, states the point plainly: a Banach space B can be isometric with its second conjugate space B** without being reflexive.[1](https://doi.org/10.1073/pnas.37.3.174)
**What quasi-reflexive means.** Every Banach space X embeds canonically in its bidual X**. Reflexive means that image is everything. Quasi-reflexive of order one means the quotient X**/X has dimension exactly one: the space misses being reflexive by a single dimension.[6](https://www.numdam.org/article/SAF_1975-1976____A20_0.pdf) James's space J has codimension one in J** under the canonical injection, and J is isomorphic to J**; an equivalent norm makes it isometrically isomorphic to J**.[3](https://math.colorado.edu/~rohi1040/expository/james_space.pdf) This combination settled two open problems at once: a Banach space with separable bidual need not be reflexive, and a separable space isomorphic to its bidual need not be reflexive.[4](https://www.impan.pl/shop/en/publication/transaction/download/product/86355) In a sequel James defined an equivalent norm on J making it isometrically isomorphic to its bidual.[4](https://www.impan.pl/shop/en/publication/transaction/download/product/86355)
The space is also structurally clean: J contains no subspaces isomorphic to c0 or ℓ1, and is not the underlying real space of any complex Banach space, since the dimension of J**/J over the reals is 1, an odd number.[3](https://math.colorado.edu/~rohi1040/expository/james_space.pdf) James himself described the discovery as accidental: he had proved a reflexivity theorem for spaces with a certain kind of basis, which lost interest when Victor Klee proved it without any basis assumption, and working through the construction produced the space.[7](https://doi.org/10.1017/cbo9780511662379.001) His explicit description of X** for such bases later gave the property M. M. Day named "shrinking".[7](https://doi.org/10.1017/cbo9780511662379.001)
## The James theorem and reflexivity criteria
James's 1964 paper *Characterizations of reflexivity* is recorded on the MaRDI research data portal, alongside related works including *A characterization of reflexive spaces*.[8](https://portal.mardi4nfdi.de/wiki/Publication:3850146) A related weak compactness theorem has since been proved in forms accessible enough to be taught in a first-year graduate functional analysis class, and continues to receive extensions and applications.[9](https://www.math.auckland.ac.nz/~moors/Survey-paper-final.pdf)
## Comparisons and downstream work
The James space sits in a family of classical counterexamples. Bessaga and Pełczyński observed that quasi-reflexivity of J implies J is not isomorphic to J ⊕ J, making it the first known infinite-dimensional Banach space not isomorphic to its Cartesian square.[4](https://www.impan.pl/shop/en/publication/transaction/download/product/86355) James's 1975–1976 seminar exposition showed the ℓ2-product of J and J* is quasi-reflexive of order two and isometric to its dual, while J is isometrically isomorphic to its bidual J** despite being non-reflexive; he also outlined a quasi-reflexive order-one space isomorphic to its dual and noted it was not known whether one could be isometric to its dual.[6](https://www.numdam.org/article/SAF_1975-1976____A20_0.pdf) • [14](https://pmc.ncbi.nlm.nih.gov/articles/PMC1063327/)
Later work built directly on the space: with an equivalent norm J becomes a semi-simple commutative Banach algebra under pointwise multiplication, with J** obtained by adjoining an identity;[10](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/on-james-quasireflexive-banach-space-as-a-banach-algebra/98784B58764FE22E78739151D8F08A42) the space has the fixed point property, is primary, and is not isomorphic to any subspace of its dual.[11](https://portal.mardi4nfdi.de/wiki/Publication:5804490) Modern constructions amalgamate it with Schreier's counterexample to the Banach–Saks property into the James–Schreier spaces, which are c0-saturated and do not embed in spaces with unconditional bases.[4](https://www.impan.pl/shop/en/publication/transaction/download/product/86355) Work continues: a 2024 arXiv paper by Spiros A. Argyros and Manuel González gives a new proof of S. Bellenot's characterization of the extreme points of the unit ball of J and an explicit description of the norm of J**.[12](https://arxiv.org/abs/2406.14104)
## By the numbers
The original PNAS note carries 141 recorded citations.[1](https://doi.org/10.1073/pnas.37.3.174)
## References
1. [A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space (PNAS, 1951), Exa library record](https://doi.org/10.1073/pnas.37.3.174)
2. [Author biography, Bases in Banach Spaces, The American Mathematical Monthly Vol. 89, No. 9 (1982)](https://www.tandfonline.com/doi/abs/10.1080/00029890.1982.11995506)
3. [Expository notes on James' space, University of Colorado](https://math.colorado.edu/~rohi1040/expository/james_space.pdf)
4. [An Amalgamation of the Banach Spaces Associated with James and Schreier, Part I, Dissertationes Mathematicae (IMPAN)](https://www.impan.pl/shop/en/publication/transaction/download/product/86355)
5. [A General Construction of Spaces of the Type of R. C. James, Canadian Journal of Mathematics](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/general-construction-of-spaces-of-the-type-of-r-c-james/C0DB90A9C18CE8B5A7B8968B37AB7339)
6. [R. C. James, Quasi-reflexive Banach spaces, Séminaire d'analyse fonctionnelle 1975–1976, exp. no. 25, Numdam](https://www.numdam.org/article/SAF_1975-1976____A20_0.pdf)
7. [Foreword by Robert C. James (retracing the discovery of J and JT), Exa library record](https://doi.org/10.1017/cbo9780511662379.001)
8. [Characterizations of reflexivity (R. C. James, 1964), MaRDI portal](https://portal.mardi4nfdi.de/wiki/Publication:3850146)
9. [A Gentle Introduction to James' Weak Compactness Theorem and Beyond, University of Auckland](https://www.math.auckland.ac.nz/~moors/Survey-paper-final.pdf)
10. [On James' Quasi-Reflexive Banach Space as a Banach Algebra, Canadian Journal of Mathematics (1980)](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/on-james-quasireflexive-banach-space-as-a-banach-algebra/98784B58764FE22E78739151D8F08A42)
11. [MaRDI portal record: A Non-Reflexive Banach Space Isometric With Its Second Conjugate Space](https://portal.mardi4nfdi.de/wiki/Publication:5804490)
12. [The Extreme Points of the Unit Ball of the James space J and its dual spaces (Argyros & González, arXiv 2024–2025)](https://arxiv.org/abs/2406.14104)
13. [id.loc.gov](https://id.loc.gov/authorities/names/n84801115.html)
14. [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC1063327/)
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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Analysts and PDE researchers › Banach space geometry specialists*
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