Packing and reflexivity in Banach spaces
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- by Clifford A. Kottman PDF
- Trans. Amer. Math. Soc. 150 (1970), 565-576 Request permission
Abstract:
A measure of the “massiveness” of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the ${l_p}$-spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to $B$-convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.References
- Mahlon M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313–317. MR 3446, DOI 10.1090/S0002-9904-1941-07451-3
- Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
- Daniel P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc. 125 (1966), 114–146. MR 205031, DOI 10.1090/S0002-9947-1966-0205031-7
- Robert C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177. MR 44024, DOI 10.1073/pnas.37.3.174
- Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
- J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578 A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3-86; English transl., Amer. Math. Soc. Transl. (2) 17 (1961), 277-364. MR 22 #2890; MR 23 #A2031. Gottfried Köthe, Topologische lineare Räume. Vol. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin and New York, 1966. MR 33 #3069.
- Vlastimil Pták, Biorthogonal systems and reflexivity of Banach spaces, Czechoslovak Math. J. 9(84) (1959), 319–326 (English, with Russian summary). MR 110008
- Robert Whitley, The size of the unit sphere, Canadian J. Math. 20 (1968), 450–455. MR 228997, DOI 10.4153/CJM-1968-041-1
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 150 (1970), 565-576
- MSC: Primary 46.10
- DOI: https://doi.org/10.1090/S0002-9947-1970-0265918-7
- MathSciNet review: 0265918