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Geometry of Banach spaces of functions associated with concave functions


Authors: Paul Hlavac and K. Sundaresan
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0388080-4
Keywords: Lorentz spaces, vector valued functions, extreme points, function spaces, rearrangement function, vector valued measures
Article copyright: © Copyright 1976 American Mathematical Society

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