The research reported in this note has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command U.S. Air Force through its European Office.

This note is based on a part of a Ph.D. thesis which was written under the supervision of Professor A Dvoretzky and Dr B Grünbaum. I wish to express my warm thanks to both for their valuable help and kind encouragement.

The boy who grew up in Israel and left it for the first time after completing his PhD never ceased to marvel at being a member of the international fellowship of mathematics.

... for many valuable discussions concerning the subject of this note.

In this work he proved theorems that exhibit remarkable, and arguably unexpected, rigidity phenomena for Banach spaces when they are deformed by uniformly continuous (or Lipschitz) mappings.

In 1965 Joram Lindenstrauss joined the Hebrew University as a new faculty, and I was a third-year undergraduate student sitting in his Banach Spaces course. This first encounter set the course of my life. The Banach space group in Jerusalem began with Dvoretzky, who was joined by his former student Grünbaum and their students. A phase transition occurred with the arrival of Joram, who joined Dvoretzky in the supervision of the doctoral theses of Lazar and Zippin, and Joram's influence dominated their work. ... I started to work with him on a master's thesis in 1966. Joram gave excellent basic and advanced courses, and within a few years the list of his students grew very fast ... Joram's supervision style was very "open." We did not have orderly weekly meetings, and he never gave me a problem for the thesis. The exposure to the different directions and problems came through his comments and criticism in the seminar and other discussions. This is also how I learned how to judge what is "important," "interesting," what is worthwhile to read, and what is publishable. His approach and views were so dominant that he was "heard" even when he was not present. Joram was very systematic and methodical. His answers to questions sounded like he had prepared a lecture on the subject. He did not like to discuss speculations: he would send me to write up my ideas and, of course, in most cases the speculation led nowhere. But when I did hand him something in writing, he was a wonderful reader. It would come back the next day with detailed feedback. He was also a wonderful writer, and his papers, books, and lecture notes (in Hebrew) are written in his typical systematic, clear, and concise style.

In the 1970s Joram's family spent several summers in Columbus (I was at Ohio State then), and my family spent one year in Jerusalem. In 1981-82 both families were in College Station, where Joram and I took our sabbaticals and Joram's wife, Naomi, worked on her PhD in computer science. Our wives became close friends and our children grew up together. Once when we went for Shabbat dinner at the Lindenstrauss home, Joram, with some help from Ayelet and Elon, built a Lego city in their living room. Our son was always happy to return to "Joram's toy store." Later I was a coach of a soccer team on which my son and Joram's daughter played, and I played basketball with Kinneret and some of her boy friends. Kinneret, now well known in the biophysics community, was the best player on the court and had a successful second career as a professional basketball player. Shabbat dinners at the Lindenstrauss home after our children were grown were particularly enjoyable when some of their children were present. It was great to get to know the adult Gallia, the youngest and most widely read of the Lindenstrauss clan (Google her to find out why), who was often present helping Naomi with the preparations.

The main purpose of these lecture notes is to give an outline of that part of Banach space theory which deals with properties of special and important classes of spaces. We have tried to present the main methods used in the theory, as well as the principal ideas involved in the proofs of the basic results. The lecture notes contain some well known and classical theorems but the main subject matter consists of recent results and research directions. Many open problems are mentioned throughout these notes.

Banach space theory has seen considerable change in recent years and these excellent lecture notes provide a highly readable and up-to-date survey of this area.

The purpose of this book is to present the main results and current research directions in the geometry of Banach spaces with an emphasis on the study of the structure of the classical Banach spaces, that is $C(K)$ and L_{p}(\mu) and related spaces. We do not attempt to write a comprehensive survey of Banach space theory, or even only of the theory of classical Banach spaces, since the amount of interesting results on the subject makes such a survey practically impossible. A part of the subject matter of this book appeared in outline in our lecture notes (1973). In contrast to those notes most of the results presented here are given with complete proofs. We therefore hope that it will be possible to use the present book as a textbook on Banach space theory and as a reference book for research workers in the area. It contains much material which was not discussed in the earlier version (1973), a large part of which is the result of very recent research work. An indication of the rapid recent progress in Banach space theory is the fact that most of the many problems stated in the earlier book have been solved by now.

A deep, original, clever and prolific mathematician, Joram Lindenstrauss ... was a pillar of modern functional analysis and an enormously influential champion of a tradition of excellence and uncompromising (at times, even harsh) pursuit of the highest possible quality of mathematical research. In addition to proving difficult theorems and solving longstanding open problems, Joram was a conceptual leader who was responsible for the formulation of foundational insights and major research directions that guided and shaped the intensive efforts of many researchers worldwide over several generations, and his legacy will undoubtedly continue to do so in the future.

My father was very honest, had high standards both for himself and for others, and always said exactly what he thought, regardless of whether what he thought was pleasant or unpleasant to hear. From the time I was a graduate student to the last talks of mine which he attended when I was already a well-established researcher, I was always especially nervous and tried to be very well prepared when giving a talk when he was in the audience, as I was sure I would be told afterwards exactly what he thought of the talk.

I recall a number of non-mathematical events, such as playing basketball on a team opposing him (he was very good), going to concerts, his daughter Gallia's pictures of Jerusalem, walks in Jerusalem, exhibitions, and of course his welcoming family and the huge amount of help we were given by his wife, Naomi.

What really set him apart from the point of view of his influence on me were the standards he set and lived by. He had very high standards as to what constitutes good mathematics and for scientific and personal integrity. These were also very easy to absorb from him: he always said what was on his mind whether asked for it or not.