Smoothness and weak sequential compactness

Authors:
James Hagler and Francis Sullivan

Journal:
Proc. Amer. Math. Soc. **78** (1980), 497-503

MSC:
Primary 46B05

MathSciNet review:
556620

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If a Banach space *E* has an equivalent smooth norm, then every bounded sequence in has a converging subsequence. Generalizations of this result are obtained.

**[1]**Edgar Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31–47. MR**0231199****[2]**Errett Bishop and R. R. Phelps,*The support functionals of a convex set*, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR**0154092****[3]**E. Cech and B. Pospisil,*Sur les espaces compacts*, Publ. Fac. Sci. Univ. Masaryk**258**(1938), 1-14.**[4]**J. Hagler and W. B. Johnson,*On Banach spaces whose dual balls are not weak* sequentially compact*, Israel J. Math.**28**(1977), no. 4, 325–330. MR**0482086****[5]**J. Hagler and E. Odell,*A Banach space not containing 𝑙₁ whose dual ball is not weak* sequentially compact*, Illinois J. Math.**22**(1978), no. 2, 290–294. MR**0482087****[6]**Richard Haydon,*On Banach spaces which contain 𝑙¹(𝜏) and types of measures on compact spaces*, Israel J. Math.**28**(1977), no. 4, 313–324. MR**0511799****[7]**K. John and V. Zizler,*On rough norms on Banach spaces*, Comment. Math. Univ. Carolin.**19**(1978), no. 2, 335–349. MR**500126****[8]**Victor Klee,*Some new results on smoothness and rotundity in normed linear spaces.*, Math. Ann.**139**(1959), 51–63 (1959). MR**0115076****[9]**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**[10]**E. B. Leach and J. H. M. Whitfield,*Differentiable functions and rough norms on Banach spaces*, Proc. Amer. Math. Soc.**33**(1972), 120–126. MR**0293394**, 10.1090/S0002-9939-1972-0293394-4**[11]**R. R. Phelps,*Support cones in Banach spaces and their applications*, Advances in Math.**13**(1974), 1–19. MR**0338741****[12]**-,*Convex functions on real Banach spaces*, unpublished lecture notes.**[13]**Haskell P. Rosenthal,*A characterization of Banach spaces containing 𝑙¹*, Proc. Nat. Acad. Sci. U.S.A.**71**(1974), 2411–2413. MR**0358307****[14]**Haskell P. Rosenthal,*Some recent discoveries in the isomorphic theory of Banach spaces*, Bull. Amer. Math. Soc.**84**(1978), no. 5, 803–831. MR**499730**, 10.1090/S0002-9904-1978-14521-2**[15]**Charles Stegall,*The Radon-Nikodým property in conjugate Banach spaces. II*, Trans. Amer. Math. Soc.**264**(1981), no. 2, 507–519. MR**603779**, 10.1090/S0002-9947-1981-0603779-1**[16]**S. L. Trojanski,*An example of a smooth space whose dual is not strictly normed*, Studia Math.**35**(1970), 305–309 (Russian). MR**0271708**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46B05

Retrieve articles in all journals with MSC: 46B05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1980-0556620-4

Article copyright:
© Copyright 1980
American Mathematical Society