Weak convergence in spaces of measures and operators

Topology and Its Applications, 2008

Glasgow Mathematical Journal, 2010

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Stochastic Processes and their Applications, 1997

We discuss three forms of convergence in distribution which are stronger than the normal weak convergence. They have the advantage that they are non-topological in nature and are inherited by discontinuous functions of the original random variables-clearly an improvement on 'normal' weak convergence. We give necessary and sufficient conditions for the three types of convergence and go on to give some applications which are very hard to prove in a more restricted setting.

Proceedings of the American Mathematical Society, 1995

This paper presents some properties of bounded linear functionals on a complete abstract M spaces, from which some criteria for weak convergence and weak compactness in such spaces are obtained.

International Journal of Mathematics and Mathematical Sciences, 2007

The main object of this paper is to introduce a new concept of weak statistically Cauchy sequence in a normed space. It is shown that in a reflexive space, weak statistically Cauchy sequences are the same as weakly statistically convergent sequences. Finally, weak statistical convergence has been discussed inlpspaces.

Pacific Journal of Mathematics, 1984

An integration theory for vector functions and operator-valued measures is outlined, and it is shown that in the setting of locally convex topological vector spaces, the dominated and bounded convergence theorems are almost equivalent to the countable additivity of the integrating measure. The measures studied are those representing the continuous linear operators on a space of continuous functions. When certain restrictions are imposed on the space involved, actual equivalence of countable additivity and the above theorems obtains, as well as equivalence of certain compactness properties of the operator being represented. An example is given which shows that, in general spaces, convergence in measure no longer implies the almost everywhere convergence of a subsequence.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1982

Journal of Mathematical Analysis and Applications, 1986

Glasgow Mathematical Journal, 2006

In this paper we present some results about wV (weak property V of Pełczyński) or property wV * (weak property V * of Pełczyński) in Banach spaces. We show that E has property wV if for any reflexive subspace F of E * , ⊥ F has property wV. It is shown that G has property wV if under some condition K w * (E * , F *) contains the dual of G. Moreover, it is proved that E * contains a copy of c 0 if and only if E contains a copy of 1 where E has property wV *. Finally, the identity between L(C(, E), F) and WP(C(, E), F) is investigated.

Glasgow Mathematical Journal, 2011

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach s...