## Classical and Modern Methods in SummabilitySummability is a mathematical topic with a long tradition and many applications in, for example, function theory, number theory, and stochastics. It was originally based on classical analytical methods, but was strongly influenced by modern functional analytical methods during the last seven decades. The present book aims to introduce the reader to the wide field of summability and its applications, and provides an overview of the most important classical and modern methods used. Part I contains a short general introduction to summability, the basic classical theory concerning mainly inclusion theorems and theorems of the Silverman-Toeplitz type, a presentation of the most important classes of summability methods, Tauberian theorems, and applications of matrix methods. The proofs in Part I are exclusively done by applying classical analytical methods. Part II is concerned with modern functional analytical methods in summability, and contains the essential functional analytical basis required in later parts of the book, topologization of sequence spaces as K- and KF-spaces, domains of matrix methods as FK-spaces and their topological structure. In this part the proofs are of functional analytical nature only. Part III of the present book deals with topics in summability and topological sequence spaces which require the combination of classical and modern methods. It covers investigations of the constistency of matrix methods and of the bounded domain of matrix methods via Saks space theory, and the presentation of some aspects in topological sequence spaces. Lecturers, graduate students, and researchers working in summability and related topics will find this book a useful introduction and reference work. |

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### Contents

CLASSICAL METHODS IN SUMMABILITY | 1 |

basic classical theory | 26 |

Special summability methods | 99 |

Tauberian theorems | 167 |

Application of matrix methods | 205 |

FUNCTIONAL ANALYTIC METHODS | 259 |

K and FKspaces | 338 |

structure of the domains | 396 |

FUNCTIONAL ANALYTIC METHODS | 457 |

Saks spaces and bounded domains | 515 |

Some aspects of topological sequence spaces | 538 |

563 | |

575 | |

### Common terms and phrases

Abel method absolutely convex apply arbitrarily given Banach space Banach-Steinhaus theorem Borel method boundedness called Cauchy sequence Cesaro methods characterization choose closed graph theorem conservative for null conservative matrix consider continuous linear conull convergence coregular Corollary countable defined Definition and Remark denotes domain dual pair example Exercise exists FK-space FK-topology following statements hold functional analytic gliding hump Hahn property implies index sequence isomorphism Kern Lemma Let X,P limit formula linear functional linear maps locally convex space locally convex topology matrix map matrix methods monotone Moreover neighbourhood basis norm notation notion null sequences obtain obviously particular pointwise Proof prove regular matrices row-finite Saks space satisfies Schur's theorem semi-normed spaces sequence spaces statements are equivalent subsets subspace summability methods Tauberian condition Tauberian theorem topological space triangle verify zero