Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smooth approximations in Banach spaces


Author: J. Vanderwerff
Journal: Proc. Amer. Math. Soc. 115 (1992), 113-120
MSC: Primary 46B20; Secondary 41A30
DOI: https://doi.org/10.1090/S0002-9939-1992-1081100-8
MathSciNet review: 1081100
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Banach space that has a locally uniformly convex (LUC) norm whose dual is also LUC is shown to admit $ {C^1}$-smooth partitions of unity. It is also established that there is a norm on a Hubert space with Lipschitz derivative that cannot be approximated uniformly on bounded sets by functions with uniformly continuous second derivative.


References [Enhancements On Off] (What's this?)

  • [1] R. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877-898. MR 0198492 (33:6647)
  • [2] K. Ciesielski and R. Pol, A weakly Lindelöf function space $ C(K)$ without any continuous injection into $ {c_0}(\Gamma )$, Bull. Polish Acad. Sci. Math. 32 (1984), 681-688. MR 786192 (86j:54031)
  • [3] R. Deville, Problèmes de renormages, J. Funct. Anal. 68 (1986), 117-129. MR 852656 (87m:46035)
  • [4] R. Deville, G. Godefroy, and V. Zizler, The three space problem for smooth partitions of unity and $ C(K)$ spaces, Math. Ann. 288 (1990), 613-625. MR 1081267 (92i:46014)
  • [5] J. Diestel, Geometry of Banach spaces-selected topics, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin-New York 1975. MR 0461094 (57:1079)
  • [6] M. Fabian, On a dual locally uniformly rotund norm on a dual Vašák space, Studia Math. (to appear). MR 1141364 (92j:46024)
  • [7] M. Fabian and S. Troyanski, A Banach space admits a locally uniformly rotund norm if its dual is a Vašák space, Israel J. Math. 69 (1990), 214-224. MR 1045374 (91e:46024)
  • [8] G. Godefroy, S. Troyanski, J. H. M. Whitfield, and V. Zizler, Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (1983), 344-352. MR 712585 (85d:46020)
  • [9] -, Locally uniformly rotund renorming and injections into $ {c_0}(\Gamma )$, Canad. Math. Bull. 27 (1984), 494-500. MR 763053 (86a:46016)
  • [10] R. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990), 261-268. MR 1041141 (91h:46045)
  • [11] -, Trees in renorming theory, preprint.
  • [12] R. Haydon and C. A. Rogers, A locally uniformly convex renorming for certain $ C(K)$, Mathematika 37 (1990), 1-8. MR 1067883 (91j:46033)
  • [13] D. McLaughlin, Smooth partitions of unity in preduals of WCG spaces, Math. Z. (to appear). MR 1184327 (93j:46020)
  • [14] A. S. Nemirovskiĭ and S. M. Semenov, On polynomial approximation of functions on Hilbert space, Mat. Sb. (NS) 21 (1973), 255-277. MR 0632033 (58:30211b)
  • [15] V. L. Šmulyan, Sur la dérivabilité de la norme dans l'espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 643-648. MR 0002704 (2:102f)
  • [16] K. Sundaresan, Geometry and nonlinear analysis in Banach spaces, Pacific J. Math. 102 (1982), 487-498. MR 686565 (84g:46028)
  • [17] M. Talagrand, Renormages de quelques $ C(K)$, Israel J. Math. 54 (1986), 327-334. MR 853457 (88d:46041)
  • [18] H. Toruńczyk, Smooth partitions of unity on some non-separable Banach spaces, Studia Math. 46 (1973), 43-51. MR 0339255 (49:4016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20, 41A30

Retrieve articles in all journals with MSC: 46B20, 41A30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1081100-8
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society