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Characterizations of elements of a double dual Banach space and their canonical reproductions

Vassiliki Farmaki (1993)

Studia Mathematica

For every element x** in the double dual of a separable Banach space X there exists the sequence ( x ( 2 n ) ) of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class B 1 ( X ) ╲ B 1 / 2 ( X ) (resp. to the class B 1 / 4 ( X ) ) as the elements with the sequence ( x ( 2 n ) ) equivalent to the usual basis of ℓ 1 (resp. as the elements with the sequence ( x ( 4 n - 2 ) - x ( 4 n ) ) equivalent to the usual basis...

Characterizations of Kurzweil-Henstock-Pettis integrable functions

L. Di Piazza, K. Musiał (2006)

Studia Mathematica

We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.

Characterizations of L 1 -predual spaces by centerable subsets

Yanzheng Duan, Bor-Luh Lin (2007)

Commentationes Mathematicae Universitatis Carolinae

In this note, we prove that a real or complex Banach space X is an L 1 -predual space if and only if every four-point subset of X is centerable. The real case sharpens Rao’s result in [Chebyshev centers and centerable sets, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2593–2598] and the complex case is closely related to the characterizations of L 1 -predual spaces by Lima [Complex Banach spaces whose duals are L 1 -spaces, Israel J. Math. 24 (1976), no. 1, 59–72].

Characterizations of spreading models of l 1

Persephone Kiriakouli (2000)

Commentationes Mathematicae Universitatis Carolinae

Rosenthal in [11] proved that if ( f k ) is a uniformly bounded sequence of real-valued functions which has no pointwise converging subsequence then ( f k ) has a subsequence which is equivalent to the unit basis of l 1 in the supremum norm. Kechris and Louveau in [6] classified the pointwise convergent sequences of continuous real-valued functions, which are defined on a compact metric space, by the aid of a countable ordinal index “ γ ”. In this paper we prove some local analogues of the above Rosenthal ’s theorem...

Characterizations of weakly compact sets and new fixed point free maps in c₀

P. N. Dowling, C. J. Lennard, B. Turett (2003)

Studia Mathematica

We give a basic sequence characterization of relative weak compactness in c₀ and we construct new examples of closed, bounded, convex subsets of c₀ failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c₀: such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

Characterizing Fréchet-Schwartz spaces via power bounded operators

Angela A. Albanese, José Bonet, Werner J. Ricker (2014)

Studia Mathematica

We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet-Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a Fréchet-Schwartz space does so in a special way. We single out this type of "rapid convergence" for a sequence...

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