Sequence spaces and nonarchimedean analysis (Q2901230)

The present book is divided into two parts. Part I concerns sequence spaces (Chapters 1--8); Part II treats non-Archimedean analysis (Chapters 9--14). NEWLINENEWLINENEWLINEPart I contains some of the main results on sequence spaces over the real and complex field obtained by the author, among others, up to until 2001. After giving in Chapter 1 some basic concepts and results on topological vector spaces, he studies in Chapters 2 and 3 Banach limits and (absolute, strongly) almost convergent sequences in \(\ell_{\infty}\). NEWLINENEWLINENEWLINEIn Chapter 4, he shows some properties of the spaces \(\ell(p)\) associated to bounded sequences \((p_k)_k\) of strictly positive real numbers, paying special attention to their Köthe-Toeplitz duals and to perfectness, i.e., when they coincide with their double Köthe-Toeplitz duals. Then the results given in Chapter 2 and 3 for \(\ell_{\infty}\) are generalized in Chapter 5 to these spaces \(\ell(p)\). Functional Banach limits and (absolute, strongly) convergent functions are treated in Chapter 6. They are continuous analogues of the corresponding discrete ones considered in previous chapters. NEWLINENEWLINENEWLINEPart I finishes by reviewing some basic facts on continuous linear transformations on Banach and Hilbert spaces (Chapter 7) and applying them to one of the most important classes of these transformations on sequence spaces, the matrix transformations (Chapter 8). NEWLINENEWLINENEWLINEPart II starts with a few basic facts on valued fields (Chapter 9), non-Archimedean normed and Banach spaces (Chapters 10 and 11, respectively) and non-Archimedean normed algebras (Chapter 12). Then the numerical range of an element of a non-Archimedean normed algebra, introduced by the author in [Indian J. Math. 24, No. 1--3, 19--24 (1982; Zbl 0527.46059)], is treated in Chapter 13. NEWLINENEWLINENEWLINEThe last chapter deals with the non-Archimedean counterpart of Chapter 8, i.e., with matrix transformations on non-Archimedean sequence spaces. The results contained in this chapter were obtained by the author in [Tamkang J. Math. 9, No. 2, 199--207 (1979; Zbl 0412.46059)] and [Bull. Inst. Math., Acad. Sin. 9, No. 4, 461--467 (1981; Zbl 0478.46061)]. The book does not provide more recent contributions (of any author) on the subject.