Let R be a subcategory of the category of all topological vector spaces. Let E be an element of R. The problem of the existence of the finest R-topology on E with the same continuous linear functionals as the original one is discussed. Remarks concerning the Hahn-Banach Extension Property are included.
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently . If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach...
We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality holds with some c independent of f iff w belongs to the well known Muckenhoupt class , and therefore iff for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....
Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...
In this paper we present a new hybrid method, called SASP method. We propose the hybridization of two methods, the simulated annealing (SA), which belong to the class of global optimization based on the principles of thermodynamics, and the descent method were we estimate the gradient using the simultaneous perturbation. This hybrid method gives better results. We use the Normal Boundary Intersection approach (NBI) based on the SASP method to solve...
Let be a boundedly -complete vector lattice. If each -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a -additive measure on the generated -field then is said to have the measure extension property. Various sufficient conditions on which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: has the...