Geometry of Banach spaces of functions associated with concave functions
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- by Paul Hlavac and K. Sundaresan
- Trans. Amer. Math. Soc. 215 (1976), 161-189
- DOI: https://doi.org/10.1090/S0002-9947-1976-0388080-4
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Abstract:
Let $(X,\Sigma ,\mu )$ be a positive measure space, and $\phi$ be a concave nondecreasing function on ${R^ + } \to {R^ + }$ with $\phi (0) = 0$. Let ${N_\phi }(R)$ be the Lorentz space associated with the function $\phi$. In this paper a complete characterization of the extreme points of the unit ball of ${N_\phi }(R)$ is provided. It is also shown that the space ${N_\phi }(R)$ is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces ${N_\phi }(R)$, and their abstract analogues ${N_\phi }(E)$, are obtained when E is a Banach space.References
- S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warsaw, 1932; reprint, Chelsea, New York, 1955. MR 17, 175.
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396 M. M. Day, Normed linear spaces, 2nd rev. ed., Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869, DOI 10.1007/978-1-4684-9440-2
- Israel Halperin, Function spaces, Canad. J. Math. 5 (1953), 273–288. MR 56195, DOI 10.4153/cjm-1953-031-3
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Richard A. Hunt, An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 (1964), 803–807. MR 169037, DOI 10.1090/S0002-9904-1964-11242-8
- Jerry A. Johnson, Extreme measurable selections, Proc. Amer. Math. Soc. 44 (1974), 107–112. MR 341068, DOI 10.1090/S0002-9939-1974-0341068-5
- E. de Jonge, A pair of mutually associated Banach function spaces, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 148–156. MR 0358330, DOI 10.1016/1385-7258(74)90005-5
- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Vypuklye funktsii i prostranstva Orlicha, Problems of Contemporary Mathematics, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0106412
- I. E. Leonard and K. Sundaresan, Smoothness and duality in $L_{p}(E,\,\mu )$, J. Math. Anal. Appl. 46 (1974), 513–522. MR 344869, DOI 10.1016/0022-247X(74)90257-1
- G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37–55. MR 33449, DOI 10.2307/1969496
- G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), 411–429. MR 44740, DOI 10.2140/pjm.1951.1.411
- G. G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127–132. MR 123182, DOI 10.1090/S0002-9939-1961-0123182-3
- George G. Lorentz and Tetsuya Shimogaki, Interpolation theorems for the pairs of spaces $(L^{p},\,L^{\infty })$ and $(L^{1},\,L^{q})$, Trans. Amer. Math. Soc. 159 (1971), 207–221. MR 380447, DOI 10.1090/S0002-9947-1971-0380447-9 W. A. J. Luxemburg, Rearrangement-invariant Banach function spaces, Queen’s Papers in Pure and Appl. Math., no. 10, Queen’s University, Kinston, Ont., 1967, pp. 83-144. W. A. Luxemburg, Riesz spaces, North-Holland, Amsterdam, 1971.
- Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
- R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6
- M. A. Rieffel, The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466–487. MR 222245, DOI 10.1090/S0002-9947-1968-0222245-2
- John V. Ryff, Orbits of $L^{1}$-functions under doubly stochastic transformations, Trans. Amer. Math. Soc. 117 (1965), 92–100. MR 209866, DOI 10.1090/S0002-9947-1965-0209866-5
- W. L. C. Sargent, Some sequence spaces related to the $l^{p}$ spaces, J. London Math. Soc. 35 (1960), 161–171. MR 116206, DOI 10.1112/jlms/s1-35.2.161
- E. M. Semenov, Embedding theorems for Banach spaces of measurable functions, Dokl. Akad. Nauk SSSR 156 (1964), 1292–1295 (Russian). MR 0173153
- M. S. Steigerwalt and A. J. White, Some function spaces related to $L_{p}$ spaces, Proc. London Math. Soc. (3) 22 (1971), 137–163. MR 279582, DOI 10.1112/plms/s3-22.1.137
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972 H. Steinhaus, Sur la probabilité de la convergence des séries, Studia Math. 2 (1930), 21-39.
- K. Sundaresan, Extreme points of the unit cell in Lebesgue-Bochner function spaces. I, Proc. Amer. Math. Soc. 23 (1969), 179–184. MR 247453, DOI 10.1090/S0002-9939-1969-0247453-2
- Kondagunta Sundaresan, Extreme points of the unit cell in Lebesgue-Bochner function spaces, Colloq. Math. 22 (1970), 111–119. MR 276753, DOI 10.4064/cm-22-1-111-119
- S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 215 (1976), 161-189
- MSC: Primary 46E40; Secondary 46B05
- DOI: https://doi.org/10.1090/S0002-9947-1976-0388080-4
- MathSciNet review: 0388080