Ivan Singer Knihy

Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces Grundlehren der mathematischen Wissenschaften

Focusing on the theory of bases in Banach spaces, this monograph consolidates over forty years of research and addresses both established results and unsolved problems. While previous functional analysis texts offer limited coverage, this work builds on earlier expository papers and includes a comprehensive bibliography. Due to the extensive nature of the subject, it is presented in two volumes, with the first volume laying the groundwork and the second volume set to explore related results and problems in greater depth.

The focus of this volume is the ongoing development of the theory of bases in Banach spaces since the publication of its first volume in 1970. It primarily includes Chapter III, reflecting significant advancements in the field while acknowledging that previous literature on functional analysis remains limited in this area. The author references earlier works and notes the minimal overlap with a recent book by Lindenstrauss and Tzafriri. Additionally, the volume highlights the emergence of survey papers that contribute to the understanding of bases in Banach spaces.

The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.