For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [19] show that any non-constant monotone property P : {0, 1}( n 2) → {0, 1} of n-vertex graphs has D(P) = Ω(n). We extend their result to 3-uniform hypergraphs. In particular, we ...

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent t...

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent t...

We characterise the weak $$^*$$ symmetric strong diameter 2 property in Lipschitz function spaces by a of its predual, Lipschitz-free space. call this new decomposable octahedrality and study duality with general. For Banach space to be decomposably octahedral it is sufficient that dual has property. Whether also necessary condition remains open.

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak * denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed po...

Recently, continuous functionals for unbounded order (norm, weak and weak*) in Banach lattices were studied. In this paper, we study the operators with respect to convergences. We first investigate approximation property of convergence. Then show some characterizations continuity uo, un, uaw uaw*-convergence. Based on these results, discuss order-weakly compact lattices.

let  and  be banach algebras, ,  and . we define an -product on  which is a strongly splitting extension of  by . we show that these products form a large class of banach algebras which contains all module extensions and triangular banach algebras. then we consider spectrum, arens regularity, amenability and weak amenability of these products.

In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of norm-to-weak∗ closedness and of property (Q) for these operators (as recently proven by Voisei). Various applications and limiting examples are given.

The paper presents quite general weak and strong laws of large numbers for weighted sums of triangular arrays of random fuzzy sets. The proofs of the results rely on an appropriate identification of random fuzzy sets with random elements in Banach spaces which satisfy a useful convexity property.

Let X be a closed subspace of a Banach space W and let F be the operator ideal of finite-rank operators. If α is a tensor norm, A is a Banach operator ideal, and λ > 0, then we call the condition “‖S‖α ≤ λ‖S‖A(X,W ) for all S ∈ F(X,X)” an inner inequality and the condition “‖T‖α ≤ λ‖T‖A(Y,W ) for all Banach spaces Y and for all T ∈ F(Y,X)” an outer inequality. We describe cases when outer inequ...