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History of Banach Spaces and Linear Operators

Albrecht Pietsch
Publication Date: 
Number of Pages: 
[Reviewed by
Kai Brunkalla
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This complete History of Banach Spaces and Linear Operators covers the topic from its infancy before Banach all the way to accomplishments in 2006. It contains three major sections. The first and most extensive collects results about Banach spaces in roughly chronological order. The second section is dedicated to the men and women who added to the theory of Banach spaces and operator theory, with a full chronology. The last section is a very extensive bibliography.

The book is a very good source for any researcher to find original work. It is far from leisure reading, since Pietsch has included many quotes in their original language without the benefit of translations. The historical part contains seven chapters, from the birth of Banach space theory to selected topics in modern Banach space theory and miscellaneous topics. Each chapter contains the major topics under investigation in the period covered, plus many links to other places in the history of Banach spaces.

Some of this material is covered in regular college courses, but rarely in chronological order and surely not in the same course. The book thus puts into perspective some seemingly separate fields of study. As such it could be used as a valuable resource to help connect topics throughout the field of mathematics.

For me the most compelling part of the book is the history of the people involved in the birth, growth and maturation of the theory of Banach spaces. Pietsch succeeds in putting a human face on these accomplishments. He brings out those that are in the darkness and puts them into the light (citing Brecht’s Threepenny Opera) and acknowledges the struggles that the people behind the mathematics had to endure.

Even though the book is very extensive it still does not cover all topics linked to Banach spaces and operator theory. In fact, the author admits this in a section called Omissions: “Even the present size does not allow the presentation to be complete.” This is reminiscent of another work  that set out to record all that is known about linear operators. Dunford and Schwartz had to give up after three volumes were published.

Overall the book is very informative on the subject it covers and a good starting point and reference for any researcher in Banach spaces and operator theory.

Kai Brunkalla teaches at Walsh University in North Canton, Ohio.

 Preface.- Acknowledgements.- Introduction.- The Birth of Banach Spaces.- Historical Roots and Basic Results.- Weak Topologies.- Classical Banach Spaces.- Basic Results from the Post-Banach Period.- Modern Banach Space Theory – Selected Topics.- Miscellaneous Topics.- Mathematics is Made by Mathematicians.- Chronology.- Quotations.- Bibliography.- Index.