Analytic norms in Orlicz spaces

Hájek, P.; Troyanski, S.

doi:10.1090/S0002-9939-00-05773-7

Analytic norms in Orlicz spaces
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by P. Hájek and S. Troyanski

Proc. Amer. Math. Soc. 129 (2001), 713-717

DOI: https://doi.org/10.1090/S0002-9939-00-05773-7

Published electronically: November 8, 2000

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Abstract:

It is shown that an Orlicz sequence space $h_M$ admits an equivalent analytic renorming if and only if it is either isomorphic to $l_{2n}$ or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent $C^\infty$-Fréchet norm, but no equivalent analytic norm.

References

R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2
Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
Robert Deville, Geometrical implications of the existence of very smooth bump functions in Banach spaces, Israel J. Math. 67 (1989), no. 1, 1–22. MR 1021357, DOI 10.1007/BF02764895
Robert Deville, Vladimir Fonf, and Petr Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998), 139–154. MR 1639743, DOI 10.1007/BF02780326
V. Fonf, One property of Lindenstrauss-Phelps spaces, Functional Anal. Appl. 13 (1979), 66–67.
Raquel Gonzalo, Upper and lower estimates in Banach sequence spaces, Comment. Math. Univ. Carolin. 36 (1995), no. 4, 641–653. MR 1378687
Raquel Gonzalo and Jesús Angel Jaramillo, Smoothness and estimates of sequences in Banach spaces, Israel J. Math. 89 (1995), no. 1-3, 321–341. MR 1324468, DOI 10.1007/BF02808207
Petr Hájek, Analytic renormings of $C(K)$ spaces, Serdica Math. J. 22 (1996), no. 1, 25–28. MR 1397698
Petr Hájek, Smooth norms that depend locally on finitely many coordinates, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3817–3821. MR 1285993, DOI 10.1090/S0002-9939-1995-1285993-3
Petr Hájek, Smooth functions on $C(K)$, Israel J. Math. 107 (1998), 237–252. MR 1658563, DOI 10.1007/BF02764011
Denny H. Leung, Some isomorphically polyhedral Orlicz sequence spaces, Israel J. Math. 87 (1994), no. 1-3, 117–128. MR 1286820, DOI 10.1007/BF02772988
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
R. P. Maleev, Norms of best smoothness in Orlicz spaces, Z. Anal. Anwendungen 12 (1993), no. 1, 123–135. MR 1239433, DOI 10.4171/ZAA/576
R. P. Maleev and S. L. Troyanski, Smooth functions in Orlicz spaces, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 355–370. MR 983394, DOI 10.1090/conm/085/983394
R. P. Maleev and S. L. Troyanski, Smooth norms in Orlicz spaces, Canad. Math. Bull. 34 (1991), no. 1, 74–82. MR 1108932, DOI 10.4153/CMB-1991-012-7
S. S. Pillai, On normal numbers, Proc. Indian Acad. Sci., Sect. A. 10 (1939), 13–15. MR 0000020
A. Pełczyński and W. Szlenk, An example of a non-shrinking basis, Rev. Roumaine Math. Pures Appl. 10 (1965), 961–966. MR 203432
J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58–62.