and Applied Analysis 3 Lemma 3. Let (X; ≽) be a Banach lattice. Then the positive coneX is weakly closed. Proof. It is clear that the positive coneX of the Banach lattice X is convex. We have mentioned that the positive coneX of the Banach latticeX is norm closed. ApplyingMazur’s lemma (see [12] or [15]), we have in a Banach space, a convex set is norm closed if and only if it is weakly closed....

Abstract. We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that l ∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner Lp(X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is...

the American Mathematical Society vol. 3 (1952) pp. 821-828. 2. Nils Sjöberg, Sur les minorantes sousharmoniques d'une fonction donnée, Proceedings of the Ninth Scandanavian Mathematical Congress, Helsingfors, 1938, pp. 309-319. 3. Marcel Brelot, Minorantes sous-harmoniques, extrémales et capacités, J. Math. Pures Appl. (9) vol. 24 (1945) pp. 1-32. 4. E. Szpilrajn, Remarques sur les fonctions s...

for any x 1 ( . . . , x,, GX. A theorem of Aolci and Rolewicz (see [18]) asserts that if in (1.3) C = 2~\ then X is p-normable. We can then equivalently re-norm X so that in (1.4) JB = 1. If in addition X is a vector lattice and ||x||<||y|| whenever |x|<|y| we say that X is a quasi-Banach lattice. As in the case of Banach lattices [13] we may make the following definitions. We shall say that X ...

The paper deals with on-line regression settings with signals belonging to a Banach lattice. Our algorithms work in a semi-online setting where all the inputs are known in advance and outcomes are unknown and given step by step. We apply the Aggregating Algorithm to construct a prediction method whose cumulative loss over all the input vectors is comparable with the cumulative loss of any linea...

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra B(E) of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c0 and `p for 1 6 p < ∞. We add a new member to this family by showing that there are exactly four closed ideals in B(E) for t...

The classical Banach-Stone theorem characterizes linear surjective isometries between C(K)-spaces. The main aim of this paper is to present a survey of Banach-Stone-theoremtype results between some function spaces. The weighted substitution operators play an important role in characterization of isometries, disjointness preserving operators, and lattice homomorphisms. Some open problems are giv...

If X is a closed subspace of a Banach space L which embeds into a Banach lattice not containing l ∞ ’s uniformly and L/X contains l ∞ ’s uniformly, then X cannot have local unconditional structure in the sense of Gordon-Lewis (GLl.u.st.). 1991 Mathematics Subject Classification. 46B03, 43A46.

A net (xα) in a vector lattice X is unbounded order convergent to x ∈ X if |xα − x| ∧ u converges to 0 in order for all u ∈ X+. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A ne...