Geometry of Banach spaces of functions associated with concave functions

Hlavac, Paul; Sundaresan, K.

doi:10.1090/S0002-9947-1976-0388080-4

Geometry of Banach spaces of functions associated with concave functions
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by Paul Hlavac and K. Sundaresan PDF

Trans. Amer. Math. Soc. 215 (1976), 161-189 Request permission

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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