洛朗·施瓦茨Laurent Schwartz

洛朗·施瓦茨出身于犹太家庭。他的父亲安塞尔姆施瓦茨（1872-1957）出生于阿尔萨斯韦斯托芬附近的巴尔布龙，时值1870-71年普法战争后不久，阿尔萨斯被德国吞并。他是法国人，不愿生活在德国，因此十四岁时离开家乡前往巴黎，成为一名外科医生。1907年，他与自己的表妹、拉比的女儿克莱尔·德布雷（1888-1972）结婚。尽管安塞尔姆在犹太信仰中长大，但他成为了无神论者，并以无神论者的方式抚养子女。他们的家族中有许多杰出人物，如克莱尔的兄弟罗伯特·德布雷教授（1882-1978），联合国儿童基金会的创始人，以及罗伯特·德布雷的儿子米歇尔·德布雷（1912-1996），他成为了一位非常成功的政治家，于1959-62年担任法国总理。这个大家族中还有杰出的数学家；雅克·阿达马娶了克莱尔·德布雷母亲的妹妹。施瓦茨是他父母三个儿子中的长子，有弟弟丹尼尔（生于1917年）和约瑟·伯特兰（生于1919年）。

施瓦茨十一岁时患上了小儿麻痹症。尽管他在几个月内康复，但这场疾病使他终生体弱。1926年9月，当他仍处于小儿麻痹症的恢复期时，他的父母在奥图耶买了一座乡间别墅。这是一座大房子，有壮丽的花园，周围是草地和田野，孩子们可以在那里玩耍。施瓦茨小时候，全家每个周末都会在奥图耶度过，但工作日住在巴黎。在他就读的巴黎中学里，施瓦茨在数学和希腊语、拉丁语等古典语言方面都很出色。他面临着一个艰难的选择，尤其是在全国‘concours général’拉丁语竞赛中名列第一、翻译名列第四之后。他必须在两种选择之间做出决定：要么在中学最后几年学习哲学和人文学科，为大学古典学专业做准备；要么学习数学和哲学。他的母亲一直在鼓励儿子学习方面发挥着重要作用，她向自己的兄弟罗伯特·德布雷征求意见。罗伯特的医学专长是儿童护理，他给出了专业意见，认为施瓦茨应该学习数学。这与中学老师们的相同意见一起，使施瓦茨决定放弃拉丁语，同时学习数学和哲学，并参加这两个科目的中学毕业会考。在让松·德·萨伊中学的最后几年，他学习了数学和哲学课程。他爱上了几何学，由一位富有启发性的老师教授，但对哲学课程感到失望，因为那里的教学要差得多。同样在让松·德·萨伊中学，他爱上了玛丽-埃莱娜·莱维，她在他所在的班级之上。玛丽-埃莱娜是保罗·莱维的女儿。

施瓦茨于1934年进入巴黎高等师范学校，在那里他受教于一些世界顶尖的数学家。1935年，他与同样学习数学的玛丽-埃莱娜订婚。与她的父亲保罗·莱维的友好关系对这位年轻人产生了重要的数学影响。保罗·莱维的主要兴趣在概率论和泛函分析，他使这位年轻人爱上了这些主题，这些主题后来成为他毕生的主要研究兴趣。大学往往是学生参与政治的时期，事实上，施瓦茨积极参与左翼信仰。他这一时期的政治活动在[1]中有所描述：-

这些年的思想激荡与政治参与相伴。尽管出身于传统的右翼背景，他却是莱昂·布鲁姆人民阵线政府的坚定支持者，直到他对该政府未能支持西班牙共和派感到失望。同样，他对共产主义的同情很快因斯大林的公审而减弱，尽管此后直到1947年，他做了十年的托洛茨基主义者。他声称从不后悔这一点，即使这几乎阻止了他前往美国领取菲尔兹奖。

在出色的本科生涯后，他于1937年毕业，获得Agrégation de Mathématiques。在这个阶段，他决定服义务兵役，而不是推迟。他被分配到D.C.A.（Défense Contre Aéronefs——防空部队），但由于身体虚弱且缺乏灵巧性，他发现自己无法完成拆卸和组装机枪等任务。他先被派往巴朗库尔，然后被派往比斯卡罗斯，尽管由于第二次世界大战他不得不延长兵役，但他从未见过战斗，并于1940年8月复员。在此期间，1938年，他与玛丽-埃莱娜结婚。事实上，他们原计划在1935年12月结婚，但玛丽-埃莱娜感染了肺结核，被送往上萨瓦省帕西的疗养院。被迫分离十八个月，期间他们只能通过信件联系，这对他们两人都极其艰难。1940年施瓦茨退伍后，他与妻子前往图卢兹，他的父母在德国入侵和法国沦陷后已搬到了那里。

在图卢兹期间，他代表巴黎高等师范学院前往那里进行口试时遇到了亨利·嘉当。事实上，Marie-Hélène也借此机会与亨利·嘉当交谈，因为她想恢复数学研究。埃利·嘉当建议他在克莱蒙费朗攻读博士学位，那里是斯特拉斯堡大学在二战初期德军入侵法国后迁往的地方。施瓦茨成为国家科学基金会（后来成为CNRS）的成员，该基金会支持他直到1942年底。在这项支持结束后，他从ARS（科学研究援助）获得资金，支持他直到战争结束。施瓦茨从驻巴黎的乔治·瓦隆那里获得数学建议。他在本科时上过乔治·瓦隆的课程Functions of a complex variable，因此早就认识他。

施瓦茨的学位论文Étude des sommes d'exponentielles Ⓣ（爱德华·斯图迪指数和），于1943年提交给斯特拉斯堡大学，其中包含以下对乔治·瓦隆所给予帮助的致谢：-

我要特别感谢乔治·瓦隆，他不仅给了我许多建议，而且通过他善意地与我进行的通信，帮助我克服了许多困难。

乔治·瓦隆是论文的审查人，夏尔·埃雷斯曼和安德烈·鲁塞尔也是。施瓦茨还在他的论文中写道：-

我还要感谢雅克·阿达马和保罗·莱维，他们指导并丰富了我的数学发展。

战争期间，他的政治活动、托洛茨基主义信仰和犹太背景使他陷入各种微妙的处境。他采用了假身份，自称施瓦茨-Marie Sélimartin，只是凭借技巧和好运的结合才逃脱了侦查。然而，他虚弱的身体状况意味着他无法协助抵抗运动。自他在巴黎高等师范学院学生时代起，他一直是托洛茨基主义政党的坚定支持者，但在战争期间他的感情开始改变[4]：-

托洛茨基主义在我于巴黎高等师范学院的那些年里给了我一种非凡的教育，显然比大多数同龄年轻人所受的教育更先进和复杂。但是，由于其思想的极端主义和宗派主义，以及其刻板的语言，它在占领期间使我变得中立。我对自己的行为以及那个时期大多数托洛茨基主义政党的行为仍然极为严厉。

1943年3月，施瓦茨的儿子Marc-André出生；这只会增加他们的危险。Marc-André后来成为一名作家和诗人，但生活悲惨——见下文。施瓦茨在1944-45学年在格勒诺布尔理学院讲学，然后搬到南锡，在那里，在让·戴尔萨特和让·迪厄多内的推荐下，他成为理学院教授。正是在他职业生涯的这个时期，他产生了关于分布论的著名工作。我们在下面更详细地描述了这个想法，但在此，我们引用施瓦茨自己对1944年提出这个想法的描述[4]：-

在我年轻时，我常常失眠数小时，从不服用安眠药。我留在床上，关灯，不写作，做数学。我的创造能量加倍，我迅速前进而不疲倦。我感到完全自由，没有日常生活和写作所施加的任何束缚。几个小时后……特别是如果出现意外困难……我会停下来，睡到早上。我会疲倦但整个第二天都快乐。……在这个特别的夜晚，我感到自信，充满兴奋感。我毫不耽搁地冲去把一切解释给亨利·嘉当，他……住在隔壁。他热情洋溢：“你瞧。你刚刚解决了微分的所有困难。现在我们再也不会有没有导数的函数了”。

他还继续在政治上活跃，在1945年法国选举中作为托洛茨基主义者参选，未能当选后，他于1946年在南锡再次参选（也未成功）。在南锡期间，他教了一批杰出的学生，包括Bernard Malgrange、雅克-路易·利翁斯、François Bruhat和亚历山大·格罗滕迪克。同样在这些年里，他成为数学界的国际明星，应哈拉尔·玻尔的邀请，于1947年10月在哥本哈根讲学，并且是1949年在温哥华举行的第一届加拿大数学大会的四位主要演讲者之一。施瓦茨的女儿Claudine于1947年出生；她成为一名数学家，是[42]的作者。（Claudine于1971年与数学家Raoul Robert结婚。）1953年，施瓦茨的妻子Marie-Hélène Schwartz因她的学位论文Formules apparentées à celles de Gauss-Bonnet et Nevanlinna-Ahlfors pour certaines applications d'une variété à n dimensions dans une autre Ⓣ（与卡尔·弗里德里希·高斯-Bonnet-罗尔夫·内万林纳和拉尔斯·阿尔福斯的某些公式相关的公式，用于n维流形到另一个的某些映射）被巴黎大学授予博士学位。乔治·瓦隆是她论文的审查人，正如十年前她丈夫的论文一样。

1953年，施瓦茨回到巴黎，成为教授，担任此职位直到1959年。他从1959年到1980年在巴黎综合理工学院任教。之后，他在巴黎第七大学度过了三年，直到1983年退休。下面我们简要介绍他卓越的数学贡献，但在我们审视这些之前，我们先叙述他在巴黎职业生涯期间参与的一些政治活动。

1956年，他是法国抗议俄罗斯入侵匈牙利的领导人之一。接着在第二年，他卷入了一件与他个人关系更为密切的事件，即阿尔及利亚的“莫里斯·奥丹事件”[1]：

莫里斯·奥丹是一位驻扎在阿尔及尔的数学家和共产主义者，正在施瓦茨的指导下撰写学位论文。但在1957年6月，这位25岁、有三个孩子且反对法国统治阿尔及利亚的人被伞兵绑架、遭受酷刑并杀害。施瓦茨不懈地呼吁正义，并在他缺席的情况下组织了一场关于这位年轻人学位论文的报告会。

他公开反对法国的军事行动，签署了著名的支持军人不服从的“121人宣言”。法国陆军部长皮埃尔·梅斯梅尔（同时也是综合理工学校的负责人）以“常识与荣誉”为由，剥夺了他在综合理工学校的职位作为回击。对此，施瓦茨回应说，既然梅斯梅尔指挥的军队认可了酷刑并提拔了施刑者，这样的言论是荒谬的。

在纽约短暂流亡后，他于两年后重新获得了职位……

施瓦茨在20世纪40年代末对数学做出的杰出贡献是他在分布理论方面的工作。上文我们给出了他对自己想出这个想法的那一晚的描述。他首次提出这些思想的出版物是Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiquesⓉ（函数、导数、约瑟夫·傅里叶变换以及数学和物理应用概念的推广），于1948年问世。分布理论是对微分和积分的一次相当大的扩展。奥利弗·赫维赛德和保罗·狄拉克曾怀着具体的应用目的对微积分进行了推广。然而，这些以及其他类似的形式计算方法并未建立在抽象而严格的数学基础之上。施瓦茨对分布理论的发展将这类方法置于可靠的基础之上，并极大地扩展了它们的应用范围，为众多领域的应用提供了强有力的工具。

他1949年在温哥华所作的讲座成为施瓦茨两卷本专著Théorie des distributions（1950，1951）的基础。尔文·席格在一篇评论中写道：

这是一部总体清晰、组织严谨且内容详尽的著作，阐述了作者提出的“分布”理论的基本方面，作者在早前的出版物中已对此进行了描述……这一理论为理论分析和应用分析中的许多常见情形提供了一种便利的形式体系，但其最重要的意义可能在于与偏微分方程的联系，尤其是双曲型方程，它对局部问题的适应性使其相较于希尔伯特空间（以及其他主要是全局性的）技巧更具优势。

在Encyclopaedia Britannica中关于“分析”的文章里，François Treves 对施瓦茨的工作描述如下：-

……施瓦茨的想法（1947年）是给出所有广义函数的统一解释，这些广义函数作为（连续）线性泛函，作用于在紧集外为零的无穷可微函数空间Cç上，从而渗透进了分析。他提供了一种系统的、严格的描述，完全基于抽象泛函分析与对偶性。值得注意的是，这样的方法已有先例，即安德烈·韦伊对局部紧群积分的表述……由于分布理论对可微性的要求，试验函数空间及其对偶空间略显复杂。这导致了对拓扑向量空间的广泛研究，超越了熟悉的大卫·希尔伯特和斯特凡·巴拿赫空间范畴，这些研究反过来又为分析本身的一些领域，如偏微分方程或多复变函数，提供了有用的新见解。施瓦茨的想法可以应用于除Cç之外的许多其他试验函数空间，正如他本人和其他人所展示的……

哈拉尔·玻尔于1950年8月30日在哈佛举行的国际数学家大会上向施瓦茨颁发了约翰·查尔斯·菲尔兹奖章，以表彰他在分布理论方面的工作。哈拉尔·玻尔[7]将施瓦茨1948年的论文描述为：-

……这篇论文无疑将作为我们时代经典的数学论文之一而流传。……我想，他引用的论文的每一位读者，都和我一样，在目睹该理论所导向的微积分整体结构的奇妙和谐，以及理解其应用对高等分析许多部分——如谱理论、potential theory，乃至整个线性偏微分方程理论——可能意味着何等本质的进步时，都会感到相当多的愉悦兴奋……

除了约翰·查尔斯·菲尔兹奖章之外，施瓦茨还获得了众多奖项、奖章和荣誉。他于1955年、1964年和1972年获得巴黎科学院颁发的奖项。1972年，他当选为科学院院士。他还获得了许多大学授予的名誉博士学位，包括洪堡大学（1960年）、布鲁塞尔大学（1962年）、隆德大学（1981年）、特拉维夫大学（1981年）、蒙特利尔大学（1985年）和雅典大学（1993年）。

施瓦茨后来在随机微分学方面的工作，他在综述文章[45]中有所描述，另见[44]。后来的政治运动包括反对美国介入越南、苏联入侵阿富汗以及俄罗斯对车臣的战争。然而，他的政治活动导致了一场家庭悲剧[21]：-

他的儿子马克-安德烈于1971年自杀，这是他在被法国民族主义者绑架后终生创伤的结果，这些民族主义者为了报复他父亲对反殖民主义的承诺和支持寻求独立的阿尔及利亚人而绑架了他。

如此参与数学和政治，人们可能会想象施瓦茨没有时间从事一项主要爱好。然而，这完全错了，因为他是一位狂热的蝴蝶收藏家，拥有超过20,000个标本。

在施瓦茨所写的其他几本书中，我们提到Méthodes mathématiques pour les sciences physiques Ⓣ（物理科学的数学方法）（1961年）。它由George Temple评论，他写道[48]：-

那些有幸看到施瓦茨教授几年前在不列颠哥伦比亚省所作讲座笔记的人会知道，他是一位清晰、精确阐述的大师，能够轻松适应数学物理学家的需求。本书的标题显然是经过仔细选择的，以表明它涉及现代数学物理中的数学方法，而不一定是数学物理学家的入门教科书。事实上，这本书确实对数学物理中的许多基本主题提供了极好的介绍，其严谨性和抽象程度可能会让英国大学的物理学生感到惊讶。... 整本书以简洁明快的风格写成，我们已经学会将这种风格与施瓦茨教授的名字联系起来。

让我们以施瓦茨的两段引文结束；第一段关于政治，第二段关于数学：-

我一直认为政治中的道德是某种本质的东西，就像感情和亲和力一样。

在数学中发现某样东西，就是克服一种抑制和一种传统。如果你不具有颠覆性，你就无法前进。

Laurent Schwartz came from a Jewish background. His father Anselme Schwartz (1872-1957) was born in Balbronn, near Westhoffen, in Alsace shortly after the Franco-Prussian War of 1870-71 which resulted in Alsace being annexed by Germany. He was French and did not like the idea of living in Germany so, at the age of fourteen he left his home town and went to Paris where he became a surgeon. He married his first cousin, Claire Debré (1888-1972), the daughter of a rabbi, in 1907. Although Anselme was brought up in the Jewish faith, he became an atheist and brought his children up as atheists. Their family contained many brilliant people such as Claire's brother, Professor Robert Debré (1882-1978), the founder of Unicef, and Robert Debré's son Michel Debré (1912-1996), who became a highly successful politician being Prime Minister of France in 1959-62. There was also leading mathematicians in the extended family; Jacques Hadamard was married to a sister of Claire Debré's mother. Laurent was the oldest of his parents three sons, having brothers Daniel (born 1917) and Bertrand (born 1919).

When he was eleven years old, Laurent contracted polio. Although he recovered in a few months, the disease left him rather weak for the whole of his life. In September 1926, when he was still in the recovery phase of polio, his parents bought a country house at Autouillet. It was a large house with magnificent gardens surrounded by meadows and fields in which the children could play. When Laurent was young, the family would spend every weekend at Autouillet but lived in Paris during the week. At the lycée he attended in Paris, Schwartz excelled at both mathematics and the classical languages of Greek and Latin. He was faced with a difficult choice, particularly after he was placed first in the national 'concours général' in Latin, and fourth in translating. He had to make a choice between spending his final school years studying philosophy and humanities to prepare for university studies in the classics or taking mathematics and philosophy. His mother, who had always played an important role in encouraging her son to study, asked her brother Robert Debré for advice. His medical expertise was in the care of children and he gave his professional opinion that Schwartz should study mathematics. This, together with the same opinions from his teachers at the lycée, led Schwartz to decide that he would drop Latin, study both mathematics and philosophy and take the baccalaureate in both subjects. In his final years at the lycée Janson de Sailly he took courses in mathematics and philosophy. He fell in love with geometry, taught by an inspiring teacher, but was disappointed by the philosophy course where the teaching was much less good. Also at the lycée Janson de Sailly he fell in love with Marie-Hélène Levy, who was in the class above him. Marie-Hélène was the daughter of Paul Lévy.

Schwartz entered the École Normale Supérieure in Paris in 1934 where he was taught by some of the leading mathematicians in the world. He became engaged to Marie-Hélène, who also studied mathematics, in 1935. Being friendly with her father, Paul Lévy, was an important mathematical influence on the young man. Lévy, whose principle interests were in probability theory and functional analysis, gave the young man a love of these topics which would become his main research interests throughout his life. University is often a time for students to become involved in politics and, indeed, Schwartz was active with left-wing beliefs. His political activities at this time are described in [1]:-

The intellectual ferment of these years was paralleled by political engagement. Though from a traditionally right-wing background, he was a strong supporter of Leon Blum's Popular Front Government until he became disillusioned by its failure to support the Spanish Republicans. Similarly, his sympathies for communism were soon dampened by Stalin's show trials, though he then spent ten years as a Trotskyite, up to 1947. He claimed never to regret this, even though it almost prevented him travelling to America to receive the Fields Medal.

He graduated, after an outstanding undergraduate career, with the Agrégation de Mathématiques in 1937. At this stage he decided that he would do his compulsory military service rather than delay it. He was assigned to the D.C.A. (Défense Contre Aéronefs - Defense Against Aircraft) but, being physically weak and lacking dexterity, he found that he was unable to perform tasks such as dismantling and assembling machine guns. He was posted first to Ballancourt, then to Biscarrosse, and although he had to extend his military service because of World War II, he never saw combat and was demobbed in August 1940. During this time, in 1938, he had married Marie-Hélène. In fact they had planned to marry in December 1935 but Marie-Hélène contracted a pulmonary tubercular infection and was sent to a sanatorium at Passy, Haut Savoie. The forced separation of eighteen months, during which time they could only correspond by letter, had been extremely difficult for both of them. After Schwartz left military service in 1940 he went with his wife to Toulouse where his parents had moved following the German invasion and the fall of France.

While in Toulouse, he met Henri Cartan when he visited there to conduct an oral on behalf of the École Normale Supérieure. In fact Marie-Hélène also took the opportunity to talk to Henri Cartan since she wanted to resume her mathematical studies. Cartan advised him to study for a doctorate at Clermont-Ferrand which is where the University of Strasbourg moved after the German armies invaded France at the start of World War II. Schwartz became a member of the Caisse National des Sciences (which later became the CNRS) which supported him until the end of 1942. After this support ended, he received funds from the ARS (Aid à la Recherche Scientifique), which supported him to the end of the war. Schwartz received mathematical advice from Georges Valiron who was based in Paris. He had known Valiron since attending his course Functions of a complex variable while he was an undergraduate.

Schwartz's thesis Étude des sommes d'exponentielles Ⓣ, submitted to the University of Strasbourg in 1943, contains the following acknowledgement as to the help that Valiron had given:-

I want to especially thank Georges Valiron who not only gave me much advice, but also, through the correspondence he kindly entered into with me, helped me to overcome many difficulties.

Valiron was an examiner of the thesis, as was Charles Ehresmann and Andre Roussel. Schwartz also wrote in his thesis:-

I would also like to express my gratitude to Jacques Hadamard and Paul Lévy who have guided and enriched my mathematical development.

During the war his political activities, Trotskyist beliefs and Jewish background put him in all manner of delicate situations. He adopted a false identity, calling himself Laurent-Marie Sélimartin, and only by a combination of skill and good luck did he escape detection. However, his weak physical condition meant that he was unable to assist the Resistance movement. He had been a staunch supporter of the Trotskyist party since his days as a student at the École Normale Supérieure but his feelings began to change during the war [4]:-

Trotskyism gave me, during my years at the ENS, a remarkable education, clearly more advanced and sophisticated than that of most youngsters of my age. But by the extremism and sectarianism of its ideas, and by its stereotyped language, it neutralised me during the occupation. My judgment remains extremely severe on my own actions as well as those of the majority of the Trotskyist party during that period.

In March 1943 the Schwartz's son Marc-André was born; this only increased their danger. Marc-André went on to become a writer and poet but had a tragic life - see below. Schwartz spent the academic year 1944-45 lecturing at the Faculty of Science at Grenoble before moving to Nancy where, on the recommendation of Jean Delsarte and Jean Dieudonné, he became a professor in the Faculty of Science. It was during this period of his career that he produced his famous work on the theory of distributions. We described this idea in more details below but, at this point, we quote Schwartz's own description of coming up with the idea in 1944 [4]:-

In my youth I used to have insomnias lasting several hours and never took sleeping pills. I remained in my bed, the light off and without writing, did mathematics. My inventive energy was redoubled and I advanced rapidly without tiring. I felt entirely free, without any of the brakes imposed by my daily life and writing. After some hours ... especially if an unexpected difficulty came up ... I would stop and sleep until morning. I would be tired but happy for the whole of the following day. ... On this particular night I felt sure of myself and filled with a sense of exaltation. I lost no time in rushing to explain everything to Henri Cartan who ... lived next door. He was enthusiastic: "There you are. You've just resolved all the difficulties of differentiation. Now we'll never again have functions without derivatives".

He also continued to be politically active, standing in the French elections of 1945 as a Trotskyist and, after failing to be elected, he again stood (also unsuccessfully) in 1946 when in Nancy. During his time in Nancy, he taught a remarkable collection of students, including Bernard Malgrange, Jacques-Louis Lions, François Bruhat, and Alexander Grothendieck. Also during these years he became an international star in the mathematical world, lecturing, at the invitation of Harald Bohr, in Copenhagen in October 1947 and was one of four main speakers at the First Canadian Mathematical Congress in Vancouver in 1949. The Schwartz's daughter, Claudine, was born in 1947; she became a mathematician and is the author of [42]. (Claudine married the mathematician Raoul Robert in 1971.) In 1953 Schwartz's wife, Marie-Hélène Schwartz, was awarded a doctorate by the University of Paris for her thesis Formules apparentées à celles de Gauss-Bonnet et Nevanlinna-Ahlfors pour certaines applications d'une variété à n dimensions dans une autre Ⓣ. Valiron was an examiner of her thesis as he had been for her husband's ten years earlier.

In 1953 Schwartz returned to Paris where he became professor, holding this position until 1959. He taught at the École Polytechnique in Paris from 1959 to 1980. He then spent three years at the University of Paris VII before he retired in 1983. We say a little below about his remarkable mathematical contributions but before we look at these we recount some of the political activity he took part in during his career in Paris.

In 1956 he was one of the leaders of protests in France against the Russian invasion of Hungary. Then in the following year he became involved in an event much closer to him personally, the "Audin Affair" in Algeria [1]:-

Audin, a mathematician and communist based in Algiers, was writing his thesis under Schwartz's supervision. But in June 1957 the 25-year-old father of three and opponent of French rule in Algeria was abducted by paratroopers, tortured and killed. Schwartz was tireless in his calls for justice, and organised a presentation of the young man's thesis in his absence.

Vocal in his opposition to the French campaign, he signed the famous "Declaration des 121" in favour of military insubordination. The riposte of Pierre Messmer, the Minister for the French Army (and, by the same token, of the École), was to strip him of his position at the Polytechnique, for reasons of "common sense and honour". To which Schwartz replied that since the Army commanded by Messmer had sanctioned torture and promoted torturers, such remarks were absurd.

After a brief exile in New York, he regained his post two years later ...

The outstanding contribution to mathematics which Schwartz made in the late 1940s was his work in the theory of distributions. Above, we gave his own description of the night he came up with the idea. The first publication in which he presented these ideas was Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques Ⓣ which appeared in 1948. The theory of distributions is a considerable broadening of the differential and integral calculus. Heaviside and Dirac had generalised the calculus with specific applications in mind. These, and other similar methods of formal calculation, were not, however, built on an abstract and rigorous mathematical foundation. Schwartz's development of the theory of distributions put methods of this type onto a sound basis, and greatly extended their range of application, providing powerful tools for applications in numerous areas.

The lectures he gave in Vancouver in 1949 became the basis for Schwartz's two-volume treatise Théorie des distributions (1950, 1951). Irving Segal writes in a review:-

This is a generally clear, carefully organized, and detailed account of the basic aspects of the theory of "distributions'' due to the author, and described by him in earlier publications ... This theory provides a convenient formalism for many common situations in theoretical and applied analysis, but its greatest significance may be in connection with partial differential equations, particularly those of hyperbolic type, where its adaptability to local problems gives it an advantage over Hilbert space (and other primarily global) techniques.

In the article on 'Analysis' in Encyclopaedia Britannica François Treves describes Schwartz's work as follows:-

... Schwartz's idea (in 1947) was to give a unified interpretation of all the generalized functions that had infiltrated analysis as (continuous) linear functionals on the space Cç of infinitely differentiable functions vanishing outside compact sets. He provided a systematic and rigorous description, entirely based on abstract functional analysis and on duality. It is noteworthy that such an approach had a precedent, in the presentation by André Weil of the integration of locally compact groups ... Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated. This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables. Schwartz's ideas can be applied to many other spaces of test-functions beside Cç, as he himself and others have shown ...

Harald Bohr presented a Fields Medal to Schwartz at the International Congress of Mathematicians in Harvard on 30 August 1950 for his work on the theory of distributions. Harald Bohr [7] described Schwartz's 1948 paper as one:-

... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ...

Schwartz has received a long list of prizes, medals and honours in addition to the Fields Medal. He received prizes from the Paris Academy of Sciences in 1955, 1964 and 1972. In 1972 he was elected a member of the Academy. He has been awarded honorary doctorates from many universities including Humboldt (1960), Brussels (1962), Lund (1981), Tel-Aviv (1981), Montreal (1985) and Athens (1993).

Later work by Schwartz on stochastic differential calculus is described by him in the survey article [45], see also [44]. Later political campaigns include those against American involvement in Vietnam, the Soviet invasion of Afghanistan, and the Russian war against Chechnya. His political activities led, however, to a family tragedy [21]:-

His son Marc-André committed suicide in 1971, the result of the life-long trauma following his kidnapping at the hands of French nationalists seeking revenge on his father for his commitment to anti-colonialism and support of the Algerians seeking independence.

With such involvement in mathematics and politics one might imagine that Schwartz would not have had time for a major hobby. This however would be entirely wrong for he was an avid collector of butterflies, with over 20,000 specimens.

Among several other books which Schwartz has written, we mention Méthodes mathématiques pour les sciences physiques Ⓣ (1961). It was reviewed by George Temple who wrote [48]:-

Those who have been privileged to see the notes of the lectures which Professor Schwartz delivered in British Columbia a few years ago will know that he is a master of clear, precise exposition which can be readily adapted to the needs of mathematical physicists. The title of this volume has clearly been chosen with some care, to indicate that it treats of mathematical methods involved in modern mathematical physics, and that it is not necessarily an introductory text-book for the mathematical physicist. The book does give, in fact, a splendid introduction to a number of basic topics in mathematical physics, treated with a degree of rigour and abstraction which may well surprise the physics student at British universities. ... The whole book is written in a concise and lucid style which we have learnt to associate with the name of Professor Schwartz.

Let us end by giving two quotes from Schwartz; the first on politics and the second on mathematics:-

I have always thought that morality in politics was something essential, just like feelings and affinities.

To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.