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Named for Banach, who was one of the great mathematicians of the twentieth century, the concept of Banach spaces figures prominently in the study of functional analysis, having applications to integral and differential equations, approximation theory, harmonic analysis, convex geometry, numerical mathematics, analytic complexity, and probability theory.
Written by a distinguished specialist in functional analysis, this book is devoted to a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference and introduction for graduate students and researchers who want to learn Banach space theory with some historical flavor.
make it look much more logical, but actually it happens quite differently. - Atiyah [2004 ] The monographs A. F. Monna: Functional Analysis in Historical Perspective (1973), and J. Dieudonne: ´ History of Functional Analysis (1981), as well as all articles devoted to the history of functional analysis deal only with the development before 1950. Now the time has come to cover the second half of the twentieth century too. I have undertaken this adventure. Let me introduce myself by telling you that I received my M. Sc. degree in 1958, just at the time when the renaissance of Banach space theory started. Thus I have ?rst-hand experience of the progress achieved during the past 50 years. Due to the explosion of knowledge, writing about functional analysis as a whole seems to be no longer possible. Hence this book is focused on Banach spaces and (abstract bounded) linear operators. Other subjects such as topologies, measures and integrals, locally convex linear spaces, Banach lattices, and Banach algebras are treated only in so far as they turn out to be relevant for this purpose. The interplay with set theory is described carefully: Which axioms are needed in order to prove the Hahn-Banach theorem? Results about non-self-adjoint operators on Hilbert spaces have been a source of inspiration for the theory of operators on Banach spaces. Such topics are discussed in great detail. However, I have omitted almost all operator-theoretic considerations that depend decisively on the existence of an inner product.
Written by a distinguished specialist in functional analysis, this book is devoted to a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference and introduction for graduate students and researchers who want to learn Banach space theory with some historical flavor.
make it look much more logical, but actually it happens quite differently. - Atiyah [2004 ] The monographs A. F. Monna: Functional Analysis in Historical Perspective (1973), and J. Dieudonne: ´ History of Functional Analysis (1981), as well as all articles devoted to the history of functional analysis deal only with the development before 1950. Now the time has come to cover the second half of the twentieth century too. I have undertaken this adventure. Let me introduce myself by telling you that I received my M. Sc. degree in 1958, just at the time when the renaissance of Banach space theory started. Thus I have ?rst-hand experience of the progress achieved during the past 50 years. Due to the explosion of knowledge, writing about functional analysis as a whole seems to be no longer possible. Hence this book is focused on Banach spaces and (abstract bounded) linear operators. Other subjects such as topologies, measures and integrals, locally convex linear spaces, Banach lattices, and Banach algebras are treated only in so far as they turn out to be relevant for this purpose. The interplay with set theory is described carefully: Which axioms are needed in order to prove the Hahn-Banach theorem? Results about non-self-adjoint operators on Hilbert spaces have been a source of inspiration for the theory of operators on Banach spaces. Such topics are discussed in great detail. However, I have omitted almost all operator-theoretic considerations that depend decisively on the existence of an inner product.
From the reviews:
"This fundamental and interesting monograph is devoted to the history (and not only) of a remarkable and rich branch of up-to-date mathematics...
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. And I think that this book will be useful to each mathematical library."-Zantralblatt Math
"Pietsch's treatment of the early stages of Banach space theory and related areas is very nice and interesting. It contains much information I was not aware of before reading this book. ... it can give the novice to the subject a good overview of what Banach space theory is about. ... The book is written in a friendly style and is very readable. ... I recommend this book to anyone interested in functional analysis ... and even to the general mathematical community." (Joram Lindenstrauss, Mathematical Reviews, Issue 2008 i)
"There is a relatively large amount of works devoted to the prehistory of functional analysis and its birth in the classical Banach period, but this monograph is exceptional in its focus on the whole 20th century. ... the author ... has successfully brought together a respectable amount of material concerning Banach spaces and linear operators. ... The book will be certainly useful not only to researchers, but also to students. ... I can warmly recommend this book 'to all who love Banach spaces.'" (Antonín Slavík, Mathematica Bohemica, Issue 2, 2010)
"This fundamental and interesting monograph is devoted to the history (and not only) of a remarkable and rich branch of up-to-date mathematics...
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. And I think that this book will be useful to each mathematical library."-Zantralblatt Math
"Pietsch's treatment of the early stages of Banach space theory and related areas is very nice and interesting. It contains much information I was not aware of before reading this book. ... it can give the novice to the subject a good overview of what Banach space theory is about. ... The book is written in a friendly style and is very readable. ... I recommend this book to anyone interested in functional analysis ... and even to the general mathematical community." (Joram Lindenstrauss, Mathematical Reviews, Issue 2008 i)
"There is a relatively large amount of works devoted to the prehistory of functional analysis and its birth in the classical Banach period, but this monograph is exceptional in its focus on the whole 20th century. ... the author ... has successfully brought together a respectable amount of material concerning Banach spaces and linear operators. ... The book will be certainly useful not only to researchers, but also to students. ... I can warmly recommend this book 'to all who love Banach spaces.'" (Antonín Slavík, Mathematica Bohemica, Issue 2, 2010)