Modern Functional Analysis in the Theory of Sequence Spaces and Matrix Transformations

Many concepts and theories in functional analysis have turned out to be powerful and widely used tools in operator theory, in particular in the theory of matrix transformations between sequences spaces in summability. We give an introduction to the basic theory of FK, BK, AK and AD spaces, the various types of dual spaces of sequence spaces, and apply the general results to the characterisations of classes of matrix operators between certain sequence spaces that arise in summability. We also study the Hausdorff measure of noncompactness and its applications to the characterisations of compact operators between sequence spaces. This is a survey paper which also includes some results of the author’s joint research with V. Rakočević and I. Djolević at the Department of Mathematics of the Faculty of Science and Mathematics at the University of Nǐs, Serbia. Although many of the results are probably known to specialists, the proofs are included for the convenience of those readers who may not be too familiar with the subject, and an appendix is added at the end containing the fundamental theorems in functional analysis in the versions they are applied.