Convergence Structures and Applications to Functional Analysis (Hardcover, 2002 ed.)

For many, modern functional analysis dates back to Banach's book [Ba32]. Here, such powerful results as the Hahn-Banach theorem, the open-mapping theorem and the uniform boundedness principle were developed in the setting of complete normed and complete metrizable spaces. When analysts realized the power and applicability of these methods, they sought to generalize the concept of a metric space and to broaden the scope of these theorems. Topological methods had been generally available since the appearance of Hausdorff's book in 1914. So it is surprising that it took so long to recognize that they could provide the means for this generalization. Indeed, the theory of topo- logical vector spaces was developed systematically only after 1950 by a great many different people, induding Bourbaki, Dieudonne, Grothendieck, Kothe, Mackey, Schwartz and Treves. The resulting body of work produced a whole new area of mathematics and generalized Banach's results. One of the great successes here was the development of the theory of distributions. While the not ion of a convergent sequence is very old, that of a convergent fil- ter dates back only to Cartan [Ca]. And while sequential convergence structures date back to Frechet [Fr], filter convergence structures are much more recent: [Ch], [Ko] and [Fi]. Initially, convergence spaces and convergence vector spaces were used by [Ko], [Wl], [Ba], [Ke64], [Ke65], [Ke74], [FB] and in particular [Bz] for topology and analysis.

General

Imprint:	Springer-Verlag New York
Country of origin:	United States
Release date:	March 2002
First published:	2002
Authors:	R. Beattie • Heinz-Peter Butzmann
Dimensions:	241 x 160 x 20mm (L x W x T)
Format:	Hardcover
Pages:	264
Edition:	2002 ed.
ISBN-13:	978-1-4020-0566-4
Categories:	Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
LSN:	1-4020-0566-0
Barcode:	9781402005664

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Review This Product

Fri, 17 Jun 2005 | Review by: Jan Harm V.

This is an excelent book on a topic that is generally not very well known. It covers the most important contributions and applications of convergence spaces to functional analysis with emphasis on the continuous convergence structure. Even though the weakening of any one of the axioms of general topology results in a significant loss of structure and content, the authors still manage to show that an interesting and rich theory can be obtained from convergence structures. An important read for all.

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