Smoothness and weak sequential compactness
Authors:
James Hagler and Francis Sullivan
Journal:
Proc. Amer. Math. Soc. 78 (1980), 497503
MSC:
Primary 46B05
MathSciNet review:
556620
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Abstract 
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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
(37 #6754)
 [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
(27 #4051)
 [3]
E. Cech and B. Pospisil, Sur les espaces compacts, Publ. Fac. Sci. Univ. Masaryk 258 (1938), 114.
 [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
(58 #2173)
 [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
(58 #2174)
 [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
(58 #23514)
 [7]
K.
John and V.
Zizler, On rough norms on Banach spaces, Comment. Math. Univ.
Carolin. 19 (1978), no. 2, 335–349. MR 500126
(80d:46028)
 [8]
Victor
Klee, Some new results on smoothness and rotundity in normed linear
spaces., Math. Ann. 139 (1959), 51–63 (1959).
MR
0115076 (22 #5879)
 [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
(80m:46017), http://dx.doi.org/10.1112/jlms/s220.1.115
 [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
(45 #2471), http://dx.doi.org/10.1090/S00029939197202933944
 [11]
R.
R. Phelps, Support cones in Banach spaces and their
applications, Advances in Math. 13 (1974),
1–19. MR
0338741 (49 #3505)
 [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, Convex functions on real Banach spaces, unpublished lecture notes.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
(50 #10773)
 [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
(80d:46023), http://dx.doi.org/10.1090/S000299041978145212
 [15]
Charles
Stegall, The RadonNikodým property in
conjugate Banach spaces. II, Trans. Amer. Math.
Soc. 264 (1981), no. 2, 507–519. MR 603779
(82k:46030), http://dx.doi.org/10.1090/S00029947198106037791
 [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
(42 #6589)
 [1]
 E. Asplund, Fréchet differentiablity of convex functions, Acta Math. 121 (1968), 3147. MR 0231199 (37:6754)
 [2]
 E. Bishop and R. R. Phelps, Support functionals of convex sets, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 2735. MR 0154092 (27:4051)
 [3]
 E. Cech and B. Pospisil, Sur les espaces compacts, Publ. Fac. Sci. Univ. Masaryk 258 (1938), 114.
 [4]
 J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not sequentially compact, Israel J. Math. 28 (1977), 325330. MR 0482086 (58:2173)
 [5]
 J. Hagler and E. Odell, A Banach space not containing whose dual ball is not sequentially compact, Illinois J. Math. 22 (1978), 290294. MR 0482087 (58:2174)
 [6]
 R. Haydon, On Banach spaces which contain and types of measures on compact spaces, Israel J. Math. 28 (1977), 313324. MR 0511799 (58:23514)
 [7]
 K. John and V. Zizler, On rough norms on Banach spaces (preprint). MR 500126 (80d:46028)
 [8]
 V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 5163. MR 0115076 (22:5879)
 [9]
 D. G. Larman and R. R. Phelps, Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc. (to appear). MR 545208 (80m:46017)
 [10]
 E. E. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120126. MR 0293394 (45:2471)
 [11]
 R. R. Phelps, Support cones in Banach spaces and their applications, Advances in Math. 13 (1974), 119. MR 0338741 (49:3505)
 [12]
 , Convex functions on real Banach spaces, unpublished lecture notes.
 [13]
 H. P. Rosenthal, A characterization of Banach spaces containing , Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 24112413. MR 0358307 (50:10773)
 [14]
 , Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803831. MR 499730 (80d:46023)
 [15]
 C. Stegall, The RadonNikodym property in conjugate Banach spaces. II (preprint). MR 603779 (82k:46030)
 [16]
 S. Troyanski, Example of a smooth space whose conjugate has no strictly convex norm, Studia Math. 35 (1970), 305309. MR 0271708 (42:6589)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005566204
PII:
S 00029939(1980)05566204
Article copyright:
© Copyright 1980
American Mathematical Society
