Matrix Methods in Analysis

On Antosik’s Lemma and the Antosik-Mikusinski Basic Matrix Theorem

“…In [1] and [2], Antosik proved two results which are called the Antosik-Mikusinski Basic Matrix Theorem and Antosik's Lemma, respectively. The theorem and lemma have been proven to be quite effective in treating various topics in Functional Analysis and Set Function Theory [1]- [6].…”

Basic matrix theorems for $\mathcal{I}$-convergence in (ℓ)-groups

“…We now turn to the second part. We know the existence of a (D)-sequence (a t,l ) t,l , satisfying (2). Thanks to super Dedekind completeness and weak σ-dis-…”

“…The classical Basic Matrix Theorem of Antosik-Mikusiński-Swartz (see [2]) was extended by P. Antosik and C. Swartz to the context of Riesz spaces (see [3]), where the so-called "( * )-convergence" is used, and was further generalized by A. Aizpuru et al (see [1]) in the case of statistical convergence, introduced in 1951 independently by H. Steinhaus and H. Fast (see [13]). Further recent studies about measures and integrals in the context of ( )-groups and Riesz spaces can be found, for example, in [7,8,9].…”

The Vitali-Hahn-Saks theorem and measuroids

“…The present note gives some versions of this classical result and of some related results. Most of the ideas in our proofs are quite standard and can be found in the beautiful treatment [3] by Diestel; see also the lecture notes [ 1 ] by Antosik-Swartz. In the sequel, let (Q, Jf) denote a fixed measurable space. We begin with a proof of the following "primitive" result.…”

“…The following is a combination of the Vitali-Hahn-Saks Theorem and the Nikodym Convergence Theorem (see [1] or [3]). …”