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**Summability through functional analysis.**
*(English)*
Zbl 0531.40008

North-Holland Mathematics Studies, 85. Amsterdam-New York-Oxford: North-Holland. XII, 318 p. $ 42.50; Dfl. 120.00 (1984).

This book is intended to serve as a source for functional analytic applications to summability. There are nineteen chapters in it. The first three chapters are devoted to matrices and Banach spaces. The Hölder, the Nörlund and other summability matrices are succinctly discussed in Chapter 2. Chapter 4 introduces FK-spaces. In Chapter 5, replaceability and consistency are presented. Chapter 6 is entitled ’bigness theorem’. Here, the conull and coregular spaces are studied. Chapter 7 deals with sequence spaces. Matrix transformations involving certain sequence spaces are given in Chapter 8. Defining functional dual, the author formulates necessary and sufficient conditions for an FK-space to contain sequence spaces such as \(c_ 0\), \(\ell^ p (p>1)\). Chapter 9 treats semiconservative spaces and matrices. Matrix extensions form the content of Chapter 11. Distinguished subspaces of FK-spaces, matrix domains and of \(c_ A\) are given in Chapters 10,12 and 13, respectively. Chapter 14 is concerned with the functional \(\mu\). The subspace \(P\) is discussed in Chapter 15. Sequential completeness and separability are considered in Chapter 16. Chapter 17, entitled ’Maps of Banach spaces’, explains maps of \(c,w\)-matrices and the Tauberian maps. Chapter 18 on algebra expatiates upon topological divisors of zero, the maximal group and the spectrum relative to the Banach algebra of conservative matrices. The book concludes with a chapter on ’Miscellany’ which contains the Toeplitz basis, applications, questions and history. There is a bibliography. An index is also given. The get-up is excellent.

The author has himself made many remarkable contributions to the subject, and his authority is evident throughout the book. This book should prove valuable as a source-material for research-workers, who may supplement the present material by reading the following books: (i) P. K. Kamthan and M. Gupta, Sequence spaces and series (1981; Zbl 0447.46002); (ii) W. H. Ruckle, Sequence spaces (1981; Zbl 0491.46007); (iii) K. Zeller and W. Beekman, Theorie der Limitierungsverfahren (1970; Zbl 0199.113).

The author has himself made many remarkable contributions to the subject, and his authority is evident throughout the book. This book should prove valuable as a source-material for research-workers, who may supplement the present material by reading the following books: (i) P. K. Kamthan and M. Gupta, Sequence spaces and series (1981; Zbl 0447.46002); (ii) W. H. Ruckle, Sequence spaces (1981; Zbl 0491.46007); (iii) K. Zeller and W. Beekman, Theorie der Limitierungsverfahren (1970; Zbl 0199.113).

Reviewer: K. Chandrasekhara Rao (Kumbakonam)

### MSC:

40H05 | Functional analytic methods in summability |

40J05 | Summability in abstract structures |

46N99 | Miscellaneous applications of functional analysis |

40-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability |

40C05 | Matrix methods for summability |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |