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History of Banach spaces and linear operators


Looking backward to the unfolding of a new mathematical branch often provides deep insights into its concepts, methods and interconnections. This is especially true for the area of Banach spaces and linear operators, topic of a new remarkable book of Albrecht Pietsch.

As Peter Zabreiko points out in his review [Zbl 1121.46002]:“This fundamental and interesting monograph is devoted to the history of a remarkable and rich branch of mathematics, the theory of (bounded) linear operators in Banach (complete normed linear) spaces. More than just a historical account, one can find here a comprehensive and deep treatment of basic concepts, methods, and results in this theory, the history of their appearance and development, and the interaction between this theory and the other branches of functional analysis and other parts of mathematics, such as set theory and mathematical logic, algebra and topology, differential and integral equations, probability theory and combinatorics, harmonic and numerical analysis, and so on. To read this book is not easy, but the material is highly intriguing and on almost every page the reader may discover something new or unexpected.”

``There are two well-known books in the history of functional analysis [A. F. Monna, `Functional analysis in historical perspective' (Oosthoek, Utrecht) (1973; Zbl 0266.46001); J. Dieudonné, `History of functional analysis' (North--Holland Math. Studies 49) (1981; Zbl 0478.46001)]. The present book is essentially distinguished from those, regarding both its volume as well as its rich contents. For one thing, its basic part is devoted to the period after the 1950s of the 20th century. Moreover, it also treats many problems which have not yet been solved. The author writes: `The book should be useful for readers who are interested in the question Why and how something happened.' I think, this is really true. I recommend this book to all who are interested in functional analysis, to beginners and researchers, all students and all professors who deal with functional analysis and its applications. Furthermore, I think that this book will be a most useful addition to any mathematical library."

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Abel prize 2010
I. M. Gelfand 1913-2009

Scientific prize winners of the ICM 2010
Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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