Investigation of double sequence spaces by soft and hard analytical methods (Q2735992)

The Ph.D.\ thesis under review is a remarkable contribution to both the theory of topological double sequence spaces and summability (for double sequences).NEWLINENEWLINEChapter 1 has an introductory character and contains basic notation and facts from the theory of locally convex spaces and also results in summability and the theory of topological sequence spaces which are candidates for an extension to double sequence spaces.NEWLINENEWLINE The author develops in Chapter 2 a `basic theory' for topological double sequence spaces and matrix maps on them. First of all, she introduces six notions \(\nu\) of convergence (\(\nu\in\{p,bp,r,e,be,c\}\)) which are among those mainly considered in the literature. For instance, a (real or complex) double sequence \(x=(x_{kl})\) is called convergent to \(a\) in the sense of Pringsheim, in short, \(p\)-convergent to \(a\), if for each \(\varepsilon>0\) there exists an \(N\in{\mathbb N}\) such that \(| x_{kl}-a| < \varepsilon\) whenever \(k,l>N\). If, in addition, \(\sup_{k,l}| x_{kl}| <\infty\), then \(x\) is said to be boundedly convergent to \(a\) in Prings\-heim's sense, in short, {bp}--convergent to \(a\). More specially, \(x\) is called convergent in the sense of Hardy or \(r\)-convergent if it is \(p\)-convergent and if all columns and rows of \(x\), considered as a matrix, are convergent. Much more generally as Pringsheim's convergence, \((x_{kl})\) is said to be \(e\)-convergent to \(a\) if for all \(\varepsilon>0\) there exists an \(l_0\in{\mathbb N}\) such that for all \(l\geq l_0\), there exists a \(k_l\in{\mathbb N}\) with \(| x_{kl}-a| \leq\varepsilon\) whenever \(k\geq k_l\). The set of all \(\nu\)-convergent double sequences is denoted by \({\mathcal C}_\nu\). It is a well-known fact that the space \(c\) of all convergent sequences, endowed with its natural topology, is a separable BK-space (i.e., Banach space with continuous coordinate functionals). Analogously, \({\mathcal C}_r\) is a separable BDK-space (Banach double sequence space with continuous coordinate functionals), whereas \({\mathcal C}_{bp}\) is a non-separable BDK-space and \({\mathcal C}_p\) and \({\mathcal C}_e\) are non-separable LFDK-spaces (i.e., the inductive limit of an increasing sequence of Fréchet double sequence spaces with continuous coordinate functionals) and not FDK-spaces. This `complicated' topological structure of \({\mathcal C}_\nu\) makes the study of the topological structure of convergence domains \({\mathcal C}_{\nu A}\) of 3- or 4-dimensional matrices much more troublesome than in the case of matrix transformation of sequences. Hence new concepts are required. For instance, the author introduces the notion of the \(\beta\)-dual of double sequence spaces \(E\) which depends on a (fixed) notion of convergence \(\nu\) for double sequence spaces, for example, on \(\nu\in \{p,bp,r,e,be,c\}\): \(\textstyle E^{\beta(\nu)} :=\Big\{u=(u_{kl})\mid\;\forall\, x=(x_{kl})\in E:\;\sum_{k,l} u_{kl}x_{kl}\text{ is }\nu\)-convergent