The product of a Gâteaux differentiability space and a separable space is a Gâteaux differentiability space

Authors:
Lixin Cheng and Marián Fabian

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3539-3541

MSC (2000):
Primary 46B20, 46G05, 26E15, 58C20

Published electronically:
July 2, 2001

MathSciNet review:
1860485

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Abstract | References | Similar Articles | Additional Information

This paper shows that the product of a Gâteaux differentiability space and a separable Banach space is again a Gâteaux differentiability space.

**[A]**Edgar Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31–47. MR**0231199****[DGZ]**Robert Deville, Gilles Godefroy, and Václav Zizler,*Smoothness and renormings in Banach spaces*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR**1211634****[F]**Marián J. Fabian,*Gâteaux differentiability of convex functions and topology*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. Weak Asplund spaces; A Wiley-Interscience Publication. MR**1461271****[G]**John R. Giles,*Convex analysis with application in the differentiation of convex functions*, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**650456****[K]**Victor Klee,*Some new results on smoothness and rotundity in normed linear spaces.*, Math. Ann.**139**(1959), 51–63 (1959). MR**0115076****[LP]**D. G. Larman and R. R. Phelps,*Gâteaux differentiability of convex functions on Banach spaces*, J. London Math. Soc. (2)**20**(1979), no. 1, 115–127. MR**545208**, 10.1112/jlms/s2-20.1.115**[NP]**I. Namioka and R. R. Phelps,*Banach spaces which are Asplund spaces*, Duke Math. J.**42**(1975), no. 4, 735–750. MR**0390721****[P]**Robert R. Phelps,*Convex functions, monotone operators and differentiability*, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR**984602**

Robert R. Phelps,*Convex functions, monotone operators and differentiability*, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR**1238715**

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Additional Information

**Lixin Cheng**

Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, People’s Republic of China

**Marián Fabian**

Affiliation:
Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06252-9

Keywords:
Convex function,
Banach space,
G\^ateaux differentiability space

Received by editor(s):
March 16, 2000

Published electronically:
July 2, 2001

Additional Notes:
The first author was supported by NSFC 10071063, F00021

The second author was supported by GA ČR 201-98-1449, GA ČR 201/01/1198, and AV 1019003

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2001
American Mathematical Society