彼得·拉克斯Peter Lax

彼得·拉克斯出生在布达佩斯的一个犹太家庭。他的母亲是Klara Kornfeld，父亲是Henry Lax，是一名医生。在拉克斯的高中学习中，数学问题解决受到特别鼓励，这确实激发了他的兴趣，就像当时许多其他有才华的匈牙利学生一样。柯尼希·德奈什是布达佩斯技术大学的教授，他为像拉克斯这样有才华的年轻人提供了大量帮助。他和罗莎·培特担任这个年轻男孩的导师。然而，在我们继续之前，我们应该试着理解导致拉克斯家族于1941年移民美国的政治事件。

匈牙利在第一次世界大战期间与德国结盟，1918年该国投降后，他们接着签署了《特里亚农条约》，使匈牙利缩小到以前面积的三分之一。三百万匈牙利人突然发现自己成了另一个国家的公民。主要是渴望收复这些领土，推动匈牙利在1936年前后走向德意联盟。反犹主义在匈牙利已经普遍存在，但与极端反犹的纳粹更紧密的联系使匈牙利犹太人的处境更加糟糕，例如拉克斯一家。犹太法律被通过，部分是为了取悦希特勒，而匈牙利似乎通过1938年和1940年的维也纳裁决实现了其领土目标，收回了一些土地。第二次世界大战于1939年开始，但匈牙利一直置身冲突之外，直到1941年它站在德国一边对俄罗斯宣战。

从某种意义上说，拉克斯一家是幸运的。Henry Lax，即拉克斯的父亲，是一位医生，他的病人中有一位是美国驻布达佩斯领事，也许更重要的是，两人是好朋友。匈牙利所走的道路对拉克斯一家来说已经很清楚，因此亨利和克拉拉通过美国领事安排移居美国。他们于1941年11月下旬离开布达佩斯，同行的还有拉克斯和他的兄弟[6]：-

... we went by train across Europe, through Germany in train compartments filled with Wehrmacht troops. We sailed for America from Lisbon on 5 December 1941. While we were on the high seas, the war broke out. So we left as immigrants and arrived in New York as enemy aliens。

果然，日本于12月7日袭击了珍珠港，英国对匈牙利宣战，匈牙利对美国宣战。抵达美国后不久，冯·诺伊曼就到拉克斯家中拜访，此前柯尼希·德奈什已告知他这位真正杰出的年轻匈牙利数学家即将到来。尽管身为“敌国侨民”，拉克斯仍能继续他的学业[6]：-

不到一个月，我和兄弟就进了高中。我去了史岱文森。我在史岱文森没有选修任何数学课程。我比大多数老师懂得还多。但我必须修英语和美国历史，我很快就爱上了美国。

三年后，即1944年，他被征入美国陆军，在德克萨斯A&M大学的一个陆军工程训练项目中度过了非常愉快的六个月，随后，他没有被运往海外作战，而是于1945年被派往洛斯阿拉莫斯，参与曼哈顿计划，建造第一颗原子弹。当然，一个来自与美国交战国家的年轻移民会参与这项绝密计划，这似乎很奇怪。然而，当时训练有素的美国科学家不够多，因此需要许多新近移民来填补空缺。

拉克斯在1945-46年间参与了洛斯阿拉莫斯科学实验室的曼哈顿计划。随后，他在1946年夏天参加了乔治·波利亚的暑期课程，之后于1947年从纽约大学获得了他的第一个学位。在攻读博士学位期间，拉克斯于1948年与Anneli Cahn结婚。他于1949年从纽约大学获得了博士学位，其学位论文为Nonlinear System of Hyperbolic Partial Differential Equations in Two Independent Variables。卡特·佛莱德里克斯曾是他的学位论文导师。之后，拉克斯返回洛斯阿拉莫斯，于1950年在科学实验室作为曼哈顿计划的工作人员工作（[9]和[10]）：-

我在洛斯阿拉莫斯度过的第一个时期，尤其是后来的接触，塑造了我的数学思维。首先，这是作为一个科学团队一部分的经历——不仅仅是数学家，还有持有不同观点的人——目标不是一个定理，而是一个产品。这是无法从书本中学到的，必须亲身参与……其次，正是在那里——那是在1950年代——我深刻认识到计算对于科学和数学的极端重要性。在冯·诺伊曼的影响下，洛斯阿拉莫斯在1950年代和1960年代初一度是计算科学领域无可争议的领导者。

拉克斯于1951年被任命为纽约大学的助理教授。他的妻子Anneli也是一位数学家，她在纽约大学学习，其博士学位由理查·科朗特指导。她于1955年因其学位论文Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics而获得博士学位。1958年，拉克斯在德国担任富布赖特讲师，同年他被提升为纽约大学的正式教授。

拉克斯在其职业生涯早期就做出了显著贡献，并继续产生改变许多数学领域方向的研究。1957年，他发表了一篇极其重要的论文Asymptotic solutions of oscillating initial value problems，其中出现了约瑟夫·傅里叶积分算子理论的雏形。当被问及这一观点为何如此新颖，使得这些想法能够获得如此广泛的应用时，拉克斯回答说（[9]和[10]）：-

这是对所发生事情的微观局部描述。它结合了对问题从大处和小处的观察。它结合了这两个方面，这赋予了它力量。微观局部观点的数值实现是通过小波和类似方法，这些方法在数值上非常强大。

当理查·科朗特于1962年提名拉克斯为国家科学院（美国）成员时，他这样描述他：-

……像少数人那样，体现了抽象数学分析与解决个别问题时最具体的力量的统一。

拉克斯在理查·科朗特数学科学研究所蓬勃发展，该研究所位于纽约大学，在那里，应用数学与相关的纯数学在令人兴奋的思想交融中一起研究，带来了巨大进展。他于1972年被任命为该研究所所长，并一直担任此职至1980年。担任这一职务的时期特别困难，因为纽约大学刚刚关闭了他们的工程学院，将该学院的数学家们并入了理查·科朗特研究所。当这些人想要建立自己的计算系，而一个新的计算机科学系刚刚成立时，这产生了摩擦。拉克斯成功地确保研究所内没有两个对立的系，但涉及的政治很困难。

2005年，拉克斯被授予极具声望的尼尔斯·阿贝尔奖。该奖授予拉克斯：-

……因其对偏微分方程的理论与应用及其解的计算做出的开创性贡献。

然而，颁奖词中对他极为杰出的成就给出了更充分的描述，我们引用其中内容，因为它对他的工作给出了特别好的总结。我们应当指出，描述拉克斯贡献的困难在于它们如此众多且重要，以至于在一篇这样长度的文章中不可能公正地对待它们。我们引用的尼尔斯·阿贝尔奖颁奖词，虽然仍然只涵盖了他工作的一部分，但仍给出了很好的说明：-

在空气动力学、气象学和弹性力学等领域中出现的方程是非线性的，并且复杂得多：它们的解可能产生奇点。想想飞机突破音障时出现的激波。20世纪50年代和60年代，拉克斯为这类非线性方程（双曲系统）的现代理论奠定了基础。他构造了显式解，识别出特别良态的方程组类，引入了重要的熵概念，并与詹姆斯·格利姆一起对解在长时间内的行为进行了深入研究。此外，他还引入了广泛使用的拉克斯-卡特·佛莱德里克斯和拉克斯-Wendroff数值格式来计算解。他在这一领域的工作对进一步的理论发展很重要。它从天气预报到飞机设计等实际应用中也极为富有成果。现代数值分析的另一个重要基石是“拉克斯等价定理”。受Richtmyer启发，拉克斯用这一定理确立了数值实现给出微分方程解的有效近似的条件。这一结果给该主题带来了极大的清晰性。……可积系统自19世纪以来一直被研究，在纯数学和应用数学中都很重要。20世纪60年代末发生了一场革命，当时马丁·克鲁斯卡尔及其合作者发现了一类新的例子，它们具有“孤子”解：在传播时保持形状的单峰波。拉克斯对这些神秘的解着迷，并找到了理解它们的统一概念，用现在称为“拉克斯对”的东西重写了方程。这发展成为整个领域的基本工具，导致了可积系统的新构造并促进了它们的研究。散射理论研究波绕过障碍物时的变化。这种现象不仅出现在流体中，例如也出现在原子物理中（埃尔温·薛定谔方程）。与Phillips一起，拉克斯发展了广泛的散射理论，并描述了解的长期行为（具体来说，能量的衰减）。他们的工作也证明在与微分方程显然相距甚远的数学领域（如数论）中很重要。这是一个不寻常且非常美丽的例子，说明为应用数学建立的框架导致了纯数学内部的新见解。

现在让我们简要地看看拉克斯写的一些书。他与Ralph S Phillips合作撰写了Scattering theory，于1967年出版。Teruo Ikebe在评论中写道：-

这是对双曲型系统时间演化的散射理论的组织良好的处理。表述清晰且富有启发性。

第二版于1989年问世，以下摘录取自作者序言：-

在这部二十多年前撰写的专著中，我们将散射理论建立在波动方程而非埃尔温·薛定谔方程之上。这一选择在当时显得古怪，但今天看来要自然得多，正如我们偏好平移表示而非谱表示一样。这一转变是由其间发现的丰富新成果带来的……一系列全新的问题源于德米特里·康斯坦丁诺维奇·法捷耶夫和Pavlov的工作。沿着伊斯拉埃尔·盖尔范德在1962年斯德哥尔摩国际数学家大会上演讲中的提示，他们证明了将拉克斯-Phillips散射理论应用于适用于双曲空间的波动方程，是自守函数理论中的一个自然工具。

事实上，拉克斯和Phillips在Scattering theory for automorphic functions（1976）中研究了自守函数的散射理论。然而，拉克斯在这两部关于散射理论的著作之间还出版了其他书。1970年，拉克斯和格利姆出版了Decay of solutions of systems of nonlinear hyperbolic conservation laws，这是一部需要熟悉两位作者早期工作的艰深著作。1972年，拉克斯与他的妻子Annelli 拉克斯和皮埃尔·萨米埃尔 Burstein一起写了Calculus with applications and computing。一位评论者写道：-

本书中的微积分内容相当标准（除了它面向应用），但计算风格是非正统的、成功的，并强烈推荐。

然而，这本本科教材在商业上并不成功。拉克斯自己说过（[9]和[10]）：-

[Anneli和我的]微积分书极其不成功，尽管其中包含许多出色的想法。部分原因是某些材料没有以学生能够吸收的方式呈现。微积分书必须精细调整，而我没有耐心去做。Anneli会有耐心，但恐怕我太欺负她了。有时我梦想重做它，因为书中的那些想法，以及我后来产生的想法，仍然有效。

SIAM于1973年在其应用数学会议系列中出版了拉克斯的Hyperbolic systems of conservation laws and the mathematical theory of shock waves。拉克斯于1997年写了Linear algebra。这本书的序言告诉我们很多关于拉克斯的思考和他对数学的方法，因此我们完整引用他自己的话：-

这本书基于一门为入学研究生设计的讲座课程，在纽约大学理查·科朗特研究所多年讲授。五十年前，线性代数作为研究课题正在退出。然而在过去五十年中，关于如何解线性方程、执行最小二乘程序、处理线性不等式系统和求矩阵特征值的新思想空前爆发。这种爆发是对越来越快的计算机和越来越大的内存所带来的机会的回应。因此，线性代数被推到了数值数学的中心舞台。这对今天该学科的教学方式产生了深远影响，部分好，部分坏。

新的数值方法的介绍给课堂带来了新鲜而令人兴奋的材料，以及现实的新应用。毕竟，许多学生上线性代数课只是为了应用。另一方面，将应用和算法置于前台掩盖了线性代数的结构——这一趋势我深感遗憾；将学生排除在埃米·诺特和埃米尔·阿廷所创造的天堂之外，对他们极为不利。本书的目标之一就是纠正这种失衡。

我写这本书的第二个目标是呈现丰富的分析结果及其一些应用：矩阵不等式、特征值估计和行列式等等。线性代数的这一美丽方面，对从事分析的工作者和物理学家如此有用，在教材中却常常被忽视。

我努力选择具有启发性、优雅且简短的证明。当有两种不同方式看待一个问题时，我喜欢把两者都呈现出来。

拉克斯最近的一本书是Functional analysis（2002），像线性代数教材一样，它源于拉克斯在理查·科朗特研究所多年讲授的研究生课程。

列出授予拉克斯的所有荣誉需要相当多的篇幅。不过我们将尽量使其相当完整。

他当选为：
Academy of Sciences（巴黎）（1982年），
National Academy of Sciences（美国）（1982年），
American Academy of Arts and Sciences（1982年），
New York Academy of Sciences（1982年），
Russian Academy of Sciences（1989年），
Hungarian Academy of Sciences（1993年），
Academia Sinica，北京（1993年），
Moscow Mathematical Society（1995年），
London Mathematical Society。

他被授予：
Lester R 莱斯特·佛特奖（1966年和1973年），
肖维勒奖（Mathematical Association of America，1974年），
诺伯特·维纳奖（American Mathematical Society和学会 of Industrial and Applied Mathematics，1975年），
National Academy of Sciences（美国）应用数学与数值科学奖（1983年），
国家科学奖章（1986年），
沃尔夫奖（沃尔夫基金会，1987年），
Leroy 凯瑟琳·斯蒂尔奖（American Mathematical Society，1992年），
尼尔斯·阿贝尔奖（挪威，2005年）。

他获得了以下机构授予的名誉学位：
肯特州立大学（1975年）、
巴黎大学（1979年）、
亚琛工业大学（1988年）、
赫瑞瓦特大学（1990年）、
特拉维夫大学（1992年）、
马里兰大学巴尔的摩分校（1993年）、
布朗大学（1993年）、
北京大学（1993年）、
德克萨斯A&M大学（2000年）。

Peter Lax was born into a Jewish family in Budapest. His mother was Klara Kornfeld and his father was Henry Lax who was a medical doctor. In Peter's high school studies, mathematical problem solving was specifically encouraged and it certainly stimulated his interest as it did for many other talented Hungarian students at this time. Denes König, who was professor at the Technical University of Budapest, did a great deal to help talented youngsters like Peter Lax. He and Rózsa Péter acted as mentors to the young boy. Before we continue, however, we should try to understand the political events which led to the Lax family emigrating to the United States in 1941.

Hungary had been aligned with Germany through World War I and after the country surrendered in 1918 they went on to sign the Treaty of Trianon which reduced Hungary to a third of its previous size. Three million Hungarians suddenly found themselves citizens of another country. It was largely a desire to recover these territories that pushed Hungary towards the German-Italian alliance around 1936. Anti-Semitism was already widespread in Hungary but closer ties with the extreme anti-Semitic Nazis made the situation much worse for Hungarian Jews like the Lax family. Jewish Laws were passed, partly to please Hitler, and Hungary seemed to be achieving its territorial objectives with the Vienna Awards of 1938 and 1940 which returned some of its lands. World War II began in 1939 but Hungary remained out of the conflict until 1941 when it entered the war against Russia on the side of Germany.

In one way the Lax family were lucky. Henry Lax, Peter's father, was a doctor who had the American consul in Budapest as one of his patients, and perhaps more importantly the two were good friends. The road that Hungary was taking was clear to the Lax family so Henry and Klara arranged through the American consul to emigrate to the United States. They left Budapest in late November of 1941 together with Peter and his brother [6]:-

... we went by train across Europe, through Germany in train compartments filled with Wehrmacht troops. We sailed for America from Lisbon on 5 December 1941. While we were on the high seas, the war broke out. So we left as immigrants and arrived in New York as enemy aliens.

Indeed Japan attacked Pearl Harbour on 7 December, Britain declared war on Hungary, and Hungary declared war on the United States. Arriving in the United States, Lax was soon visited in his home by von Neumann who had been told by Denes König that this really outstanding young Hungarian mathematician was coming. Despite being an 'enemy alien' Lax was able to continue his education [6]:-

Within a month, my brother and I were in high school. I went to Stuyvesant. I didn't take any mathematics courses at Stuyvesant. I knew more than most of the teachers. But I had to take English and American history, and I quickly fell in love with America.

Three years later, in 1944, he was drafted into the United States Army and spent six very pleasant months at Texas A&M at an Army training programme in engineering then, instead of being shipped overseas to fight, he was sent to Los Alamos in 1945 to participate in the Manhattan Project building the first atomic bomb. Of course, it seems strange that a young immigrant from a country at war with the United States would participate in this top-secret project. However, there were not enough highly trained American scientists and so many recent immigrants were needed to fill the gap.

Lax was involved in the Los Alamos Scientific Laboratory Manhattan Project during 1945-46. He then took a summer course with Pólya in the summer of 1946 before obtaining his first degree from New York University in 1947. While studying for his doctorate, Lax married Anneli Cahn in 1948. He received his PhD in 1949, also from New York University, for his thesis Nonlinear System of Hyperbolic Partial Differential Equations in Two Independent Variables. Kurt Friedrichs had been his thesis advisor. Lax then returned to Los Alamos to spend 1950 working at the Scientific Laboratory as a Staff Member on the Manhattan Project ([9] and [10]):-

The first time I spent in Los Alamos, and especially the later exposure, shaped my mathematical thinking. First of all, it was the experience of being part of a scientific team - not just of mathematicians, but people with different outlooks - with the aim being not a theorem, but a product. One cannot learn that from books, one must be a participant ... Secondly, it was there - that was in the 1950s - that I became imbued with the utter importance of computing for science and mathematics. Los Alamos, under the influence of von Neumann, was for a while in the 1950s and the early 1960s the undisputed leader in computational science.

Lax was appointed as an Assistant Professor at New York University 1951. His wife Anneli was also a mathematician and she studied at New York University where her doctorate was supervised by Courant. She was awarded a PhD in 1955 for her thesis Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics. In 1958 Lax was a Fulbright Lecturer in Germany and in the same year he was promoted to full professor at New York University.

Lax had made remarkable contributions early in his career and he continued to produce research which changed the direction of many areas of mathematics. In 1957 he published an extremely important paper Asymptotic solutions of oscillating initial value problems where the beginnings of the theory of Fourier integral operators appeared. Asked what was so novel about the viewpoint that made the ideas able to enjoy such wide application, Lax replied ([9] and [10]):-

It is a micro-local description of what is going on. It combines looking at the problem in the large and in the small. It combines both aspects, and that gives it its strengths. The numerical implementation of the micro-local point of view is by wavelets and similar approaches, which are very powerful numerically.

When Courant nominated Lax for membership of the National Academy of Sciences (United States) in 1962 he described him as:-

... embodying, as few others do, the unity of abstract mathematical analysis with the most concrete power in solving individual problems.

Lax thrived in the Courant Institute of Mathematical Sciences New York University where applied mathematics was studied alongside relevant pure mathematics in an exciting mix of ideas which led to great progress. He was appointed director of the Institute in 1972, continuing in this role until 1980. It was a particularly difficult time to take on this role since New York University had just closed down their School of Engineering, moving the mathematicians from that School into the Courant Institute. This produced friction when these people wanted to set up their own computing department while a new Computer Science Department had just been founded. Lax succeeded in ensuring that there were not two rival departments in the Institute, but the politics involved was difficult.

In 2005 Lax was awarded the highly prestigious Abel Prize. The Prize was awarded to Lax:-

... for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.

However, a much fuller description of his quite outstanding achievements was given in the citation and we quote from this as it gives a particularly good summary of his work. We should remark that the difficulty in giving a description of Lax's contributions is that they are so numerous and important that in an article of this length it is impossible to do them justice. The Abel Prize citation which we quote, while still only covering a part of his work, still gives a good indication:-

The equations that arise in such fields as aerodynamics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities. Think of the shock waves that appear when an airplane breaks the sound barrier. In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems). He constructed explicit solutions, identified classes of especially well-behaved systems, introduced an important notion of entropy, and, with Glimm, made a penetrating study of how solutions behave over a long period of time. In addition, he introduced the widely used Lax-Friedrichs and Lax-Wendroff numerical schemes for computing solutions. His work in this area was important for the further theoretical developments. It has also been extraordinarily fruitful for practical applications, from weather prediction to airplane design. Another important cornerstone of modern numerical analysis is the 'Lax Equivalence Theorem'. Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation. This result brought enormous clarity to the subject. ... Integrable systems have been studied since the 19th century and are important in pure as well as applied mathematics. In the late 1960s a revolution occurred when Kruskal and co-workers discovered a new family of examples, which have "soliton" solutions: single-crested waves that maintain their shape as they travel. Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called "Lax pairs". This developed into an essential tool for the whole field, leading to new constructions of integrable systems and facilitating their study. Scattering theory is concerned with the change in a wave as it goes around an obstacle. This phenomenon occurs not only for fluids, but also, for instance, in atomic physics (Schrödinger equation). Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy). Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory. This is an unusual and very beautiful example of a framework built for applied mathematics leading to new insights within pure mathematics.

Let us now look briefly at some books which Lax wrote. He collaborated with Ralph S Phillips in writing Scattering theory published in 1967. Teruo Ikebe wrote in a review:-

This is a well-organized treatment of scattering theory for the time evolution of systems of hyperbolic type. The presentation is clear and instructive.

A second edition appeared in 1989 and the following extract is taken from the authors Preface:-

In this monograph, written more than twenty years ago, we based our scattering theory on the wave equation rather than the Schrödinger equation. That choice seemed eccentric then but appears much more natural today, as does our preference for the translation representation over the spectral representation. This change was brought about by a wealth of new results discovered in the intervening years ... An entirely new set of problems originated in the work of Faddeev and Pavlov. Following up on a hint in Gelfand's address to the 1962 Stockholm International Congress, they showed that the Lax-Phillips scattering theory, applied to the wave equation appropriate to hyperbolic space, is a natural tool in the theory of automorphic functions.

In fact the use of scattering theory for automorphic functions was studied by Lax and Phillips in Scattering theory for automorphic functions (1976). However, Lax published other books between these two texts on scattering theory. In 1970 Lax and Glimm published Decay of solutions of systems of nonlinear hyperbolic conservation laws, a difficult work which requires familiarity with earlier work of both authors. In 1972 Lax, together with his wife Annelli Lax and Samuel Burstein, wrote Calculus with applications and computing. A reviewer wrote:-

The calculus material in this book is fairly standard (except that it is oriented towards applications) but the computing flavour is unorthodox, successful and highly recommended.

However, this undergraduate text was not a great commercial success. Lax himself said ([9] and [10]):-

[Anneli and my] calculus book was enormously unsuccessful, in spite of containing many excellent ideas. Part of the reason was that certain materials were not presented in a fashion that students could absorb. A calculus book has to be fine-tuned, and I didn't have the patience for it. Anneli would have had it, but I bullied her too much, I am afraid. Sometimes I dream of redoing it because the ideas that were in there, and that I have had since, are still valid.

SIAM published Lax's Hyperbolic systems of conservation laws and the mathematical theory of shock waves in their Conference Series in Applied Mathematics in 1973. Lax wrote Linear algebra in 1997. The Preface to this book tells us much about Lax's thinking and his approach to mathematics so we quote fully his own words:-

This book is based on a lecture course designed for entering graduate students and given over a number of years at the Courant Institute of New York University. Fifty years ago, linear algebra was on its way out as a subject for research. Yet during the past five decades there has been an unprecedented outburst of new ideas about how to solve linear equations, carry out least square procedures, tackle systems of linear inequalities, and find eigenvalues of matrices. This outburst came in response to the opportunity created by the availability of ever faster computers with ever larger memories. Thus, linear algebra was thrust centre stage in numerical mathematics. This had a profound effect, partly good, partly bad, on how the subject is taught today.

The presentation of new numerical methods brought fresh and exciting material, as well as realistic new applications, to the classroom. Many students, after all, are in a linear algebra class only for the applications. On the other hand, bringing applications and algorithms to the foreground has obscured the structure of linear algebra - a trend I deplore; it does students a great disservice to exclude them from the paradise created by Emmy Noether and Emil Artin. One of the aims of this book is to redress this imbalance.

My second aim in writing this book is to present a rich selection of analytical results and some of their applications: matrix inequalities, estimates for eigenvalues and determinants, and so on. This beautiful aspect of linear algebra, so useful for working analysts and physicists, is often neglected in texts.

I strove to choose proofs that are revealing, elegant, and short. When there are two different ways of viewing a problem, I like to present both.

A fairly recent book by Lax is Functional analysis (2002) which, like the linear algebra text, grew out of graduate lectures that Lax gave at the Courant Institute over many years.

To list all the honours that have been given to Lax takes up quite a bit of space. However we shall try to make it fairly complete.

He was elected to :
the Academy of Sciences (Paris) (1982),
the National Academy of Sciences (United States) (1982),
the American Academy of Arts and Sciences (1982),
the New York Academy of Sciences (1982),
the Russian Academy of Sciences (1989),
the Hungarian Academy of Sciences (1993),
the Academia Sinica, Beijing (1993),
the Moscow Mathematical Society (1995),
the London Mathematical Society.

He was awarded:
the Lester R Ford Award (1966 and 1973),
the Chauvenet Prize (Mathematical Association of America, 1974),
the Norbert Wiener Prize (American Mathematical Society and Society of Industrial and Applied Mathematics, 1975),
the National Academy of Sciences (United States) Award in Applied Mathematics and Numerical Sciences (1983),
the National Medal of Science (1986),
the Wolf Prize (The Wolf Foundation, 1987),
the Leroy Steele Prize (American Mathematical Society, 1992),
the Abel Prize (Norway, 2005).

He was awarded honorary degrees by:
Kent State University (1975),
the University of Paris (1979),
RWTH Aachen (1988),
Heriot-Watt University (1990),
Tel Aviv University (1992),
the University of Maryland, Baltimore (1993),
Brown University (1993),
Beijing University (1993),
Texas A&M University (2000).