里斯Frigyes Riesz

里斯是Ignácz 里斯（1843-1918）的儿子，Ignácz 里斯是一名医生，母亲是Szidónia Nagel（1858-1930）。也许在我们开始这本传记之前，我们应该稍微谈谈里斯这个名字。这个名字是匈牙利语，但当匈牙利是奥匈帝国的一部分时，许多匈牙利人被赋予德语名字，后来他们更改这些名字以显示他们的匈牙利身份。里斯实际上使用了他名字的几种变体，以里斯、Frédéric、Friedrich或Frederick的名字发表作品，这些分别是匈牙利语、法语、德语和英语版本。我们将在本传记中始终使用里斯这个名字。

里斯的父亲Ignácz 里斯毕业于杰尔的本笃会中学，然后在维也纳学习（1861-1866）以获得医学文凭。他从1866年起住在杰尔，在那里他担任杰尔犹太学校董事会副主席（1873-1906），并在经济上支持社区的小学。在1903年和1909年，他是杰尔犹太人在匈牙利犹太人第12区会议上的代表之一。他是杰尔歌唱和音乐协会的成员。Ignácz 里斯于1878年9月15日在杰尔与Szidónia Nagel结婚。Szidónia Nagel是商人Benő Nagel和他的妻子Róza的女儿。Benő Nagel从1860年代起住在杰尔，在那里他是一名粮食商人并创办了一家银行。他还在宗教社区中担任职务（教堂建设委员会、司库），并从1874年到1891年担任杰尔西盖特（杰尔的一个区）联合宗教社区的主席。

Ignácz和Szidónia 里斯住在杰尔历史市中心Kazinczy街和Jedlik Ányos街拐角处的一所房子里。他们有六个孩子：里斯（生于1880年），本传记的主题；Dezső 里斯（生于1881年），Isabell 里斯（生于1883年）；马塞尔·里斯（生于1886年）；Sándor 里斯（生于1888年）；和Margit 里斯（生于1894年）。让我们稍微谈谈里斯的兄弟姐妹。Dezső 里斯于1881年12月31日出生在杰尔，仅几天后于1882年1月3日去世。除了她的出生年份，我们没有关于Isabell 里斯的进一步信息，但相信她也在婴儿期去世。马塞尔·里斯于1886年11月16日出生在杰尔，成为著名数学家，并在本档案中有传记。Sándor 里斯于1888年2月29日出生在杰尔。他在布达佩斯大学法学院学习，并于1915年成为克卢日人民援助办公室主任。他在第一次世界大战期间在东线服役并受伤。1919年底，他在布达佩斯开设了一家律师事务所，然后从1947年起，他成为塞格德的公证人。Margit 里斯于1894年12月3日出生在杰尔。她嫁给了德语和法语教师Alfréd Szauer。1944年，Margit、Alfréd和他们的女儿Zsuzsa成为奥斯维辛大屠杀的受害者。

尽管里斯家族是犹太人，里斯和他的兄弟姐妹都出生在犹太教信仰中，但里斯的中等教育是在杰尔本笃会高中接受的。他于1889年九岁时开始在那里学习。他受到数学和物理教师的影响，其方式在[19]中有所描述：-

里斯在杰尔上中学，Dániel Arany在那里任教。当Dániel Arany创办《中学数学杂志》时，里斯14岁。这份杂志对他来说出现得正是时候。然而两年后，Dániel Arany离开了教职，由一位雄心勃勃的年轻教师Zoltán Kovács接替，后者在大学就读时深受厄特沃什·罗兰的影响，偏爱自然科学而非定量科学。他很快开始编写教科书，不久他的教科书《高中物理》就在全国多所中学使用。我们不知道这位新物理教师和一位对数学感兴趣的才华横溢的学生之间可能发生了什么，但事实是，1896/97年发表在Lapok上的物理问题解答样本大多是由里斯寄来的。里斯中学毕业后进入苏黎世联邦理工学院，甚至从那里向Lapok寄送解答，这绝非巧合。

我们还应注意，在杰尔本笃会高中就读期间，里斯不仅在数学方面表现出色，在中学数学竞赛中获奖，而且在写作方面也很出色，在七年级（1895-1896）和八年级（1896-1897）的文学竞赛中都获奖。他于1897年从杰尔本笃会高中毕业，同年晚些时候在苏黎世的苏黎世联邦理工学院开始大学学习。他在苏黎世联邦理工学院学习了两年数学和物理，但认定自己对数学的兴趣大于物理，并且也以教师为职业目标，于是于1899年转学到布达佩斯。在布达佩斯大学，他修读了朱利叶斯·科尼格、屈尔沙克·约瑟夫和Manó Beke（1862-1946）开设的数学课程，但对理论物理保持兴趣，听了厄特沃什·罗兰和Izidor 阿尔布雷希特·弗勒利希（1852-1931）的讲座。里斯在久拉·瓦利的指导下攻读博士学位，并于1902年凭借学位论文A negyedrendű elsőfajú térgörbén lévő pontkonfigurációk helyzetgeometriai tárgyalása Ⓣ（一阶空间曲线上点构形的位置几何）获得学位。他的引言开头如下：-

继阿尔弗雷德·克莱布什在三阶平面曲线理论方面的基础性工作之后，卡尔·古斯塔夫·阿克塞尔·哈纳克等人将第一类四阶空间曲线的点的齐次坐标构造为参数的不同椭圆函数；这些构造的共同性质是，曲线与任意代数曲面的交点由以下关系刻画：它们的参数之和是周期的。这一关系将曲线构形的研究归结为数论同余式的研究，由此得出的结论比几何结论简单得多、透明得多；事实上，关于这些构形的大部分结果都是由分析研究提供的；分析方法很快建立起一整套定理体系，而几何方法几乎无法为其提供基础。看来这一体系现已通过分析研究完成；至少没有希望再得到任何重要的、实质性的新结果。如果我仍然从这样一个经过修正的范围内来衡量我的主题，我这样做是因为这项任务只是表面上不讨好：因为曲线点构形的理论远未完成。数学家负有双重任务：寻找事实，并研究所发现的事实与已知事实之间的联系；将所发现的纳入一个无矛盾的连贯体系。我们以分析方式发现了大部分结果，因为这样更容易发明；但根据我们理论主题的性质，几何，或者更严格地说，不带任何度量性质的研究，属于位置几何的框架。

获得博士学位后，里斯继续在布达佩斯大学学习，以取得中学教师资格证书，并于1903年获得数学和物理两科的证书。

里斯渴望了解更多关于德国哥廷根大学正在研究的最新数学发展。他从1903年11月2日到1904年4月30日在哥廷根，参加了大卫·希尔伯特的讲座。回到匈牙利后，他开始自愿服兵役，之后于1904年8月30日被任命为莱沃恰国立科学院代课教师。1906年9月，他成为莱沃恰学院的正式教师。里斯出生在一个犹太家庭，在犹太教信仰中长大，但在1906年，他皈依了归正会，这是匈牙利最大的新教基督教会。[14]的作者们想知道他的工作转为正式与受洗加入基督教会之间是否有联系。这是一个有趣的问题，我们怀疑我们永远也不会知道答案。

尽管在莱沃恰担任教师，里斯仍在产出重要的数学成果。他是泛函分析的奠基人之一，其工作对物理学有许多重要应用。他在学位论文中基于莫里斯·弗雷歇引入的思想，利用莫里斯·弗雷歇的距离概念，将昂利·勒贝格关于实函数的工作与大卫·希尔伯特及其学生艾尔哈德·施密特发展的积分方程领域联系起来。里斯给出了平方昂利·勒贝格可积函数的表示定理，实质上表明这类函数构成一个完备度量空间。该论文于1907年发表在科学院的Comptes Rendus上，比Ernst Fischer在同一期刊上发表类似结果早两个月。这一结果现在称为里斯-Fischer定理，是昂利·勒贝格积分理论的伟大成就之一。我们注意到，里斯-Fischer定理是大卫·希尔伯特空间的约瑟夫·傅里叶分析中的基本定理。它是证明矩阵力学与波动力学等价性的数学基础。这在早期量子理论中具有根本重要性。

1908年4月，里斯出席了在罗马举行的国际数学家大会，并于4月7日在第1分会作了题为Stetigkeitsbegriff und abstrakte Mengenlehre Ⓣ（连续性与抽象集合论的概念）的报告。他的报告开头如下：-

我首先解释今天在这里所说的连续性是什么意思。为了有一个参照点，我提到那些关于几何基础的研究，它们将连续性，或者更确切地说，连续延拓流形的概念，置于其假设的首位。最著名的是波恩哈德·黎曼、索菲斯·李和大卫·希尔伯特的研究。在波恩哈德·黎曼的工作中，连续延拓流形的概念还有些模糊；在索菲斯·李的工作中，至少就其被使用的程度而言，它被定义到这样的程度，以至于它隐含在问题的分析表述中。然而，关键的一点——即这种概念化主要是对极限元素的定义，或者更一般地说，是对凝聚点的定义——只有在大卫·希尔伯特的工作中才足够清晰地显现出来。凝聚点概念的定义在那里是通过假设一种将所考虑的结构映射到某些数流形上的可能性来实现的，这种映射也满足某些指定条件；然而，凝聚点的概念对于那些数流形来说已经被定义了。

Béla Szőkefalvi-Nagy 在25中写道：-

在1908年于罗马举行的国际数学家大会上，里斯提出了拓扑空间的公理，直接对极限点概念进行公理化，并由此得到了在现代拓扑学中占据确定地位的拓扑空间类，即$T_{1}$-空间类。因此，科学应当感谢里斯首次成功引入了拓扑空间的概念。

里斯出席了在罗马举行的国际数学家大会，他的地址仍在Levoča（他给出了该名称的匈牙利语版本，Lőcse），但那年晚些时候他搬到了布达佩斯，当时他被任命为布达佩斯第三区一所Gimnázium的教师。

在1909年的一篇论文中，里斯得出了与1907年那篇类似的结果，但用的是汤姆斯·斯蒂尔吉斯积分。第二年他引入了$q$重昂利·勒贝格可积函数空间，从而开始了赋范函数空间的研究，因为对于$q ≥ 3$，这类空间不是大卫·希尔伯特空间。里斯引入了函数序列$(f_{n}(x) )$的“弱收敛”概念。正交函数级数的令人满意的理论只有在昂利·勒贝格积分发明之后才成为可能，而这一理论主要是里斯的工作。他1910年的工作标志着算子理论的开端。

在布达佩斯中学任教期间，里斯致力于他的教授资格论文（Habilitation）论文，以便能被任命到布达佩斯大学的一个大学职位。然而，在完成他的任教资格论文之前，1911年他申请了布达佩斯大学人文学院第三数学系的一个职位。他是申请该职位的六人之一，审查候选人的委员会在推荐中将里斯排在第一位。大学对委员会提出的推荐进行了多次投票，并将职位提供给了József Suták（1865-1954），他是佩斯Piarist中学的一名教师。在里斯未能被任命到布达佩斯大学后不久，1911年10月5日，科洛兹堡大学（该城市现在称为Cluj-Napoca）数学与自然科学学院一致同意邀请他填补高等定量科学系正教授的空缺职位。该部于1911年10月20日批准了这一决定，因此里斯停止了他在布达佩斯大学的任教资格论文工作。事实上，他于1912年4月6日被任命为科洛兹堡大学的特聘教授，并于1914年4月6日被任命为正教授。他继续他的创新研究，1918年他的工作接近于斯特凡·巴拿赫空间的公理化理论，而这一理论两年后由斯特凡·巴拿赫在其学位论文中公理化地建立起来。

里斯被任命到Cluj，该城市当时在匈牙利境内，以Kolozsvár之名著称。第一次世界大战对匈牙利不利，1918年战争结束后，周边国家的军队入侵了该国。1919年5月，里斯写了一封信（几乎可以肯定是写给戈弗雷·哈罗德·哈代的），解释了他和同事们所遭受的异常困难[20]：-

负责占领该城的罗马尼亚军队，连同我们那些希望将特兰西瓦尼亚（因而也是匈牙利很大一部分）与罗马尼亚王国统一的罗马尼亚族同胞，切断了我们与外界的一切通讯手段。因此，在过去三个月里，我完全没有收到任何消息，既没有在斯德哥尔摩Hogskola任教的兄弟的消息，也没有住在距此500公里的匈牙利西部的母亲和姐妹的消息，她们一定在这该死的布尔什维克运动中遭受着残酷的苦难。我已经数月没有邮件了。没有报纸，除了一些罗马尼亚报纸，甚至连科学期刊也没有！它们必须在邮局被销毁，这个空缺以后将很难填补！但与最近的事件相比，这只是一件蠢事。本月10日，罗马尼亚人在军事力量的支持下，宣布我们的大学财产属于罗马尼亚国家。教授们被要求宣誓效忠罗马尼亚国家及其国王。由于我们——依据国际法的规定——一致拒绝这种对我们祖国的背叛，大学在本月12日，即他们提出要求48小时后，在上课期间遭到突然包围，他们动用军事力量，教授们被逐出研究所，我们的科学设备被拿走，大约2000名学生因所有大学事务立即暂停而被驱散！

匈牙利政府被迫于1920年6月4日签署《特里亚农条约》。匈牙利只剩下先前匈牙利不到三分之一的土地。罗马尼亚、捷克斯洛伐克和南斯拉夫都接管了大片地区，但奥地利、波兰和意大利也从匈牙利获得了土地。《特里亚农条约》签订后，Kolozsvár不再属于匈牙利，而是属于罗马尼亚，并更名为Cluj，因此那里的匈牙利大学不得不迁往新的匈牙利边界内，并于1920年迁至Szeged，那里此前没有大学。

1922年在塞格德，里斯与阿尔弗雷德·哈尔合作建立了鲍耶数学研究所。当然，该研究所以这位著名匈牙利数学家的名字命名，他的出生地是科洛兹堡，而大学刚刚被迫从该镇迁出。里斯成为该研究所新创办的期刊Acta Scientiarum Mathematicarum的编辑，该期刊迅速成为数学的重要来源。他在这本期刊上发表了许多论文，1922年的第一篇是关于德米特里·叶戈罗夫的线性泛函定理。它发表在第一卷的第一部分。自1922年11月27日起，他担任塞格德国家中学教师资格考试委员会的副主席。他还被选为1925-1926学年的自然科学学院院长[25]：-

在1925年作为大学校长所作的就职演讲中，他以“高等数学中的初等方法”为题，里斯用高等数学的例子表明，在科学中，“高等”、“复杂”和“困难”并非永久的形容词，今天仍然如此的事物明天可能变得“初等”、“简单”和“容易”。

1928年，里斯参加了在意大利博洛尼亚举行的国际数学家大会，在那里他主持了I-B分会的一场会议，并在I-C分会作了题为Sur la décomposition des opérations fonctionnelles linéaires Ⓣ（论线性泛函运算的分解）的演讲。Béla Szőkefalvi-Nagy在[25]中写道：-

在1928年博洛尼亚国际数学家大会的演讲中，里斯指出，连续函数的线性运算的相关部分可以在不使用汤姆斯·斯蒂尔吉斯积分的情况下被刻画和生成。他的方法基于优运算的概念。他的基本定理是，任何线性运算的优集总是有一个最小的优。他的方法具有完全一般性的巨大优势，因此它不仅适用于连续函数的线性运算，也适用于在任意抽象集合上解释的函数。

当然，尽管里斯在塞格德很快乐，但他还是希望被任命到布达佩斯大学。1936年出现了一个机会，当时József Suták，他在1911年优先于里斯被任命，已从布达佩斯Pázmány 罗莎·培特大学的高等几何系退休。大学没有公开招聘这个职位，而是费耶尔被学院要求向他们推荐最合适的候选人。费耶尔列出了21位匈牙利数学家，有些在匈牙利工作，有些在国外，他们都有资格。在他的报告中，他写道里斯是独一无二的：-

……即使在优秀者中，也有一位数学家脱颖而出，如果我们审视他纯数学发现的份量及其经受时间考验的性质，以及他所有优点的众多之处，目前无人能及。

在写了七页赞扬里斯杰出品质的文字后，费耶尔向系里提议不经申请就邀请里斯到高等几何系。系里于1936年11月26日开会，未经辩论就以11票赞成、29票反对否决了这一提议。没有给出拒绝的理由，但很难看出理由除了里斯的犹太出身之外还能是什么。

1944年3月，德国军队进入匈牙利。里斯受邀于1944年春季在日内瓦大学作两场讲座，但由于他当时面临的问题而无法出席。我们注意到，他未能亲自发表的两场讲座的文本以Sur la théorie ergodique Ⓣ（论遍历理论）（1945）为题发表。在塞格德，里斯被要求佩戴黄星，被迫退休，并被告知将被限制在隔都中。瑞典大使馆请求允许他受到特殊待遇，经内政部长批准，他获准留在自己的公寓中。他缝上了黄星，但总是在外面穿一件大衣。他于1944年8月1日卸任教授职位。尽管遭受了屈辱，他活了下来。他的兄弟Sándor 里斯也活了下来，但正如我们上面提到的，他的妹妹Margit与她的丈夫和女儿一起在奥斯维辛被杀害。

Géza 伊姆雷·拉卡托斯于1944年8月被任命为匈牙利总理，以图使匈牙利与纳粹德国保持距离，并与盟国寻求单独媾和。新政府迅速推翻了针对犹太人的决定，里斯被任命为塞格德大学1944-1945学年第二学期的校长。1945年底，里斯被任命为布达佩斯大学数学讲席。他的兄弟马塞尔·里斯当时住在瑞典，为里斯组织了在瑞典和丹麦大学为期三个月的巡回讲座，从1947年12月到1948年3月，他进行了这次巡回讲座。

里斯在泛函分析中的许多基本发现与斯特凡·巴拿赫的发现结合在一起。他对其他领域也做出了许多贡献，包括遍历理论，他在1938年给出了平均遍历定理的一个初等证明。他还研究了正交级数和拓扑学。Rogosinski在22中写到里斯的风格：-

F 里斯的工作不仅因其成果的真正重要性而杰出，也因他在数学品味和措辞上的审美辨别力而杰出。……F 里斯风格的更为从容的大师风范，无论他用母语匈牙利语写作，还是用法语或德语写作，都传达出这样的愉悦，并且对较年长的数学家来说，是我们有失去危险的东西的怀旧残余。对他来说，没有仅仅为了结构理论而存在的抽象，他总是回到某些具体而实质的情境中的应用。

他的书Leçon's d'analyse fonctionnelleⓉ（《泛函分析教程》）（1952）是有史以来写得最易读的泛函分析著作之一。Rogosinski这样描述这本由里斯与其学生Béla Szökefalvi-Nagy合写的书[22]：-

在这里，在他自己写的前半部分中，我们看到这位老大师向我们描绘他所看到的实分析，充满爱意，从容不迫，并带着艺术家的辨别眼光。我毫不怀疑，这本书将作为数学文献宝库中的经典而留存。有了它，以及他的所有其他工作，Frederic 里斯作为一位伟大而多产的数学家的记忆，将在我们这门艺术的历史中长久留存。

关于这本书的详情，包括序言和评论的摘录，见THIS LINK。

关于那些认识里斯的人所提供的细节，见THIS LINK。

里斯因其工作获得了许多荣誉。1917年，他获得了匈牙利科学院颁发的托莫里·阿纳斯塔兹基金会奖。他于1915年当选为匈牙利科学院的通讯院士，并于1936年当选为正式院士。推荐他成为正式院士的人包括费耶尔和Béla Kerékjártó，他们明确指出，在他成为通讯院士后的二十年里：-

……他早期的发现产生了如此深远和广泛的影响，在此期间他为许多新的研究方向铺平了道路，这些方向已经引发了最活跃的科学运动，而且里斯现在被全世界的数学家公认为一流的领军数学家。……里斯-Fischer定理已经有了许多显著的应用，但现在已表明，在理论物理中，维尔纳·海森堡量子力学等价于埃尔温·薛定谔量子力学，而旧的里斯定理为证明这一非常重要的事实提供了基础。……作为一名研究者、教师和期刊编辑，里斯将自己的全部力量奉献给为他的国家服务，他为自己是其最杰出的科学家之一而感到自豪。

Hungarian Academy of Sciences于1927年授予他马尔齐巴尼奖，并于1946年授予他大奖。1949年，他被授予金级科苏特奖，1953年又被授予科苏特大奖。1950年，他获得了人民共和国功勋勋章。他于1948年当选为巴黎科学院的通讯院士，并于1954年当选为巴伐利亚科学院的外籍院士。他还当选为瑞典隆德皇家地理学会的成员。他于1946年获得塞格德大学、1950年获得布达佩斯大学、1954年获得巴黎索邦大学的荣誉博士学位。

里斯的泛函分析书是他的最后一部出版物，因为他的健康开始恶化。他生命的最后六个月在Kútvölgyi疗养院度过，并于1956年2月在那里去世。在归正会举行了一场小型家庭葬礼后，他被安葬在Kerepesi公墓（其正式名称为Fiume路国家公墓）。我们注意到，厄特沃什·罗兰和费耶尔也安葬在Kerepesi公墓。

Frigyes Riesz was the son of Ignácz Riesz (1843-1918), who was a medical man, and Szidónia Nagel (1858-1930). Perhaps before we begin this biography we should say a little about the name Frigyes. This name is Hungarian but, when Hungary was part of the Austria-Hungarian Empire, many Hungarians were given German names which they later changed to show their Hungarian identity. Frigyes Riesz actually used several variants of his name, publishing under the names Frigyes, Frédéric, Friedrich or Frederick, these being the Hungarian, French, German and English versions respectively. We will use the name Frigyes throughout this biography.

Frigyes' father, Ignácz Riesz, graduated from the Benedictine Gymnasium in Győr, and then studied in Vienna (1861-1866) for his medical diploma. He lived in Győr from 1866 where he was vice-president of the Győr Jewish school board (1873-1906) and he financially supported the community's elementary school. In 1903 and 1909, he was one of the representatives of the Győr Jews at the 12th district meeting of the Hungarian Jews. He was a member of the Győr Singing and Music Association. Ignácz Riesz married Szidónia Nagel on 15 September 1878 in Győr. Szidónia Nagel was the daughter of the merchant Benő Nagel and his wife Róza. Benő Nagel lived in Győr from the 1860s where he was a grain merchant and founded a bank. He also held positions in the religious community (church building committee, treasurer), and from 1874 to 1891 he was the president of the united religious community of Győrsziget, a district of Győr.

Ignácz and Szidónia Riesz lived in a house on the corner of Kazinczy Street and Jedlik Ányos Street, located in the historic city center of Győr. They had six children: Frigyes Riesz (born 1880), the subject of this biography; Dezső Riesz (born 1881), Isabell Riesz (born 1883); Marcel Riesz (born 1886); Sándor Riesz (born 1888); and Margit Riesz (born 1894). Let us say a little about Frigyes's siblings. Dezső Riesz was born in Győr on 31 December 1881 and died only days later on 3 January 1882. Other than the year of her birth, we have no further information about Isabell Riesz but believe she also died as a baby. Marcel Riesz, born in Győr on 16 November 1886, became a famous mathematician and has a biography in this archive. Sándor Riesz was born in Győr on 29 February 1888. He studied in the Faculty of Law of the University of Budapest and became head of the Cluj People's Aid Office in 1915. He served on the Eastern Front during World War I and was wounded. At the end of 1919 he opened a law office in Budapest then, from 1947, he was a notary in Szeged. Margit Riesz was born in Győr on 3 December 1894. She married the German and French language teacher Alfréd Szauer. In 1944, Margit, Alfréd and their daughter Zsuzsa became victims of the Holocaust in Auschwitz.

Although the Riesz family were Jewish and Frigyes and his siblings were all born into the Jewish faith, Frigyes' secondary education was at the Győr Benedictine High School. He began his studies there in 1889 when he was nine years old. He was influenced by teachers of mathematics and physics in the way described in [19]:-

Frigyes Riesz attended high school in Győr, where Dániel Arany taught. When Dániel Arany started the Secondary School Mathematical Journals, Frigyes Riesz was 14 years old. The journal started at the best moment for him. Two years later, however, Dániel Arany left the faculty, and was replaced by an ambitious, young teacher, Zoltán Kovács, who had graduated from university under the spell of Loránd Eötvös and preferred natural sciences to quantitative sciences. He soon began writing textbooks, and soon his textbook "Physics for the Upper Secondary School Classes" was used in various secondary schools across the country. We do not know the details of the process that may have taken place between the new physics teacher and a talented student interested in mathematics, but it is a fact that most of the sample solutions to physics problems posted in Lapok in 1896/97 were sent in by Frigyes Riesz. It cannot be a coincidence that after graduating from high school, Riesz enrolled at the Zurich University of Technology and even sent solutions to Lapok from there.

We should also note that while at the Győr Benedictine High School Frigyes Riesz not only excelled in mathematics, winning prizes in the Secondary School Mathematics Competition, but he also excelled in writing, winning prizes in literary competitions in both his seventh grade (1895-1896) and his eighth grade (1896-1897). He graduated from the Győr Benedictine High School in 1897 and later that year began his university studies in Zurich at the Eidgenössische Technische Hochschule. For two years he studied mathematics and physics at the ETH in Zurich but, deciding that he was more interested in mathematics than physics and also aiming as a career as a teacher, he transferred to Budapest in 1899. At the University of Budapest he took mathematics courses given by Gyula König, József Kürschák and Manó Beke (1862-1946), but maintained an interest in theoretical physics attending lecture by Loránd Eötvös and Izidor Fröhlich (1852-1931). Riesz studied for his doctorate advised by Gyula Vályi and was awarded the degree in 1902 for his thesis A negyedrendű elsőfajú térgörbén lévő pontkonfigurációk helyzetgeometriai tárgyalása Ⓣ. His introduction begins:-

Following the fundamental work of Clebsch in the theory of plane curves of the third order, Harnack and others have constructed the homogeneous coordinates of the points of a fourth order space-curve of the first kind as different elliptic functions of a parameter; the common property of these constructions is that the intersections of the curve with an arbitrary algebraic surface are characterised by the relation that the sum of their parameters is periodic. This relation, reducing the investigation of the configurations of the curve to the investigation of number-theoretical congruences, has led to conclusions that are considerably simpler and more transparent than geometric conclusions; and indeed, the greater part of the results concerning these configurations have been provided by analytical investigations; the analytical method has soon built up a whole system of theorems for which the geometric method could scarcely provide the basis. It seems that this system has now been completed by means of analytical investigations; at least there is no prospect of any important, substantially new results. If I nevertheless measure my subject from such a revised circle, I do so because the task is only seemingly thankless: for the theory of point configurations of a curve is by no means finished. The mathematician has the double task of searching for facts and investigating the connection between the facts found and those already known; of inserting the found into a coherent system without contradiction. We have found the greater part of the results analytically, because invention is easier in this way; but by the nature of the subject of our theory, geometry, or, with even greater restriction, the investigations devoid of all metric character, belongs to the framework of positional geometry.

After the award of his doctorate, Riesz continued to study at the university of Budapest for his secondary school teaching certificate and he was awarded this for both mathematics and physics in 1903.

Riesz was keen to learn more about the latest mathematical developments which were being studied at Göttingen University in Germany. He spent from 2 November 1903 to 30 April 1904 at Göttingen where he attended lectures by David Hilbert. Returning to Hungary, he began voluntary military service before being appointed as a substitute teacher at the Levoča State Academy of Sciences on 30 August 1904. In September 1906 he became a full-time teacher at the Levoča Academy. Riesz had been born into a Jewish family and brought up in the Jewish faith but, in 1906, he converted to the Reformed Church, the largest Protestant Christian church in Hungary. The authors of [14] wonder if there is a connection between his job becoming full-time and his baptism into the Christian Church. It is an interesting question to which we doubt we will ever know the answer.

Although working as a teacher in Levoča, Riesz was producing important mathematical results. He was a founder of functional analysis and his work has many important applications in physics. He built on ideas introduced by Fréchet in his dissertation, using Fréchet's ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt. Riesz produced a representation theorem for quadratic Lebesgue integrable functions essentially showing that the space of such functions is a complete metric space. The paper was published in Comptes Rendus of the Academy of Sciences in 1907, two months before a similar result by Ernst Fischer was published in the same journal. The result, now called the Riesz-Fischer theorem, is one of the great achievements of the Lebesgue theory of integration. We note that the Riesz-Fischer theorem is fundamental in the Fourier analysis of Hilbert space. It was the mathematical basis for proving that matrix mechanics and wave mechanics were equivalent. This is of fundamental importance in early quantum theory.

In April 1908 Riesz attended the International Congress of Mathematicians in Rome and gave the talk Stetigkeitsbegriff und abstrakte Mengenlehre Ⓣ in Section 1 on 7 April. He begins his talk as follows:-

I begin by explaining in what sense I am speaking of continuity here today. To have a point of reference, I refer to those investigations into the foundations of geometry that place continuity, or more precisely, the concept of a continuously extended manifold, at the forefront of their assumptions. The best known are those of Riemann, Lie, and Hilbert. In Riemann's work, the concept of a continuously extended manifold is still somewhat vague; in Lie's, at least to the extent that it is used, it is defined to such an extent that it is implicitly contained in the analytical formulation of the problem. However, the essential point - namely, that this conceptualisation is primarily a definition of the limiting element, or more generally, a definition of the point of condensation - only emerges with sufficient clarity in Hilbert's work. The definition of the concept of the condensation point is achieved there by postulating a possibility of mapping the structures under consideration onto certain manifolds of numbers, which mapping also fulfils certain specified conditions; however, the concept of the condensation point is already defined for those manifolds of numbers.

Béla Szőkefalvi-Nagy writes in [25]:-

At the International Mathematical Congress in Rome held in 1908, Riesz formulated the axioms of a topological space, directly axiomatizing the concept of limit point and in this way arrived at the class of topological spaces which took a definitive place in modern topology under the name of the class of $T_{1}$-spaces. Thus, science can thank Frigyes Riesz for the first successful introduction of the concept of topological space.

When Riesz attended the International Congress of Mathematicians in Rome his address was still in Levoča (he gives the Hungarian version of the name, Lőcse) but later that year he moved to Budapest when he was appointed as a teacher at a Gimnázium in the 3rd district of Budapest.

In a 1909 paper, Riesz produced a similar result to his 1907 one but in terms of a Stieltjes integral. The following year he introduced the space of $q$-fold Lebesgue integrable functions and so he began the study of normed function spaces, since, for $q ≥ 3$ such spaces are not Hilbert spaces. Riesz introduced the idea of the 'weak convergence' of a sequence of functions $(f_{n}(x) )$. A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz. His work of 1910 marks the start of operator theory.

While teaching at the Gimnázium in Budapest, Riesz worked on his habilitation thesis so that he might be appointed to a university position in the University of Budapest. Before completing his habilitation thesis, however, in 1911 he applied for a position in the Third Mathematics Department of the Faculty of Humanities of the University of Budapest. He was one of six who applied for this post and the committee examining the candidates ranked Riesz first in their recommendations. The University took several votes on the recommendations made by the committee and offered the position to József Suták (1865-1954), a teacher at the Piarist Gymnasium in Pest. Shortly after Riesz failed to be appointed to the University of Budapest, on 5 October 1911 the Faculty of Mathematics and Natural Sciences of the University of Kolozsvár (the city is now known as Cluj-Napoca) unanimously agreed to invite him to fill the vacant position as full professor in the Department of Higher Quantitative Sciences. The ministry approved the decision on 20 October 1911 and, as a consequence Riesz stopped work on his University of Budapest habilitation thesis. In fact he was appointed as an extraordinary professor in Kolozsvár on 6 April 1912 and a full professor on 6 April 1914. He continued his innovative research and in 1918 his work came close to an axiomatic theory for Banach spaces, which were set up axiomatically two years later by Banach in his dissertation.

When Riesz was appointed to Cluj the city was in Hungary and known under the name Kolozsvár. World War I went badly for Hungary and, after the war ended in 1918 armies from surrounding countries invaded the country. In May 1919 Riesz wrote a letter (almost certainly to G H Hardy) explaining the extraordinary difficulties he and his colleagues were suffering [20]:-

The Romanian army in charge of occupying the city, together with those of our own compatriots of Romanian nationality wishing the unification of Transylvania (and therefore of a great part of Hungary) with the Romanian Kingdom, have cut us off from all means of communication with the rest of the world. Therefore, for the last three months I have not had any news at all either from my brother, who teaches in the Hogskola in Stockholm, nor from my mother and my sister, who live in western Hungary, 500 kilometres away from here, and who must be suffering cruelly under this cursed Bolshevistic movement. I have been deprived of mail for months. No newspapers, except for some Romanian newspapers, not even scientific journals! They have to be destroyed at the post office, a gap that will be very difficult to fill later! But this is only a silliness compared with recent events. On the 10th of this month, the Romanians, supported by military force, have declared that our university property belongs to the Romanian state.The professors have been asked to swear an oath of loyalty to the Romanian state and its king. And as we - leaning on the prescriptions of international law - unanimously declined such a betrayal of our fatherland, the university was beleaguered unexpectedly on the 12th of this month, 48 hours after the delivery of their request, during classes, using military force, the professors were banished from their institutes, our scientific equipment was taken, and about 2000 students were dispersed as a consequence of the immediate suspension of all university affairs!

The Hungarian government was forced to sign the Treaty of Trianon on 4 June, 1920. Hungary was left with less than one third of the land that had previously been Hungary. Romania, Czechoslovakia and Yugoslavia all took over large areas but Austria, Poland and Italy also gained land from Hungary. Kolozsvár was no longer in Hungary after the Treaty of Trianon but rather it was in Romania and was renamed Cluj, so the Hungarian University there had to move within the new Hungarian borders and it moved to Szeged in 1920, where there had previously been no university.

In Szeged in 1922 Riesz set up the János Bolyai Mathematical Institute in a joint venture with Alfréd Haar. Of course the Institute was named after the famous Hungarian mathematician whose birthplace was Kolozsvár, the town from which the university had just been forced to move. Riesz became editor of the newly founded journal of the Institute, the Acta Scientiarum Mathematicarum, which quickly became a major source of mathematics. He was to publish many papers in this journal, the first in 1922 being on Egorov's theorem on linear functionals. It was published in the first part of the first volume. He was the vice-president of the National Secondary School Teacher Examination Board in Szeged from 27 November 1922. He was also elected Rector of the Faculty of Natural Sciences for the academic year 1925-1926 [25]:-

In his inaugural speech as university rector in 1925, which he gave under the title "Elementary Methods in Higher Mathematics", Frigyes Riesz demonstrated, using examples from higher mathematics, that in science "higher", "complicated", and "difficult" are not permanent adjectives, and that what is still such today may become "elementary", "simple", and "easy" tomorrow.

In 1928 Riesz attended the International Congress of Mathematicians in Bologna, Italy where he chaired one of the Section I-B sessions and gave the lecture Sur la décomposition des opérations fonctionnelles linéaires Ⓣ in Section I-C. Béla Szőkefalvi-Nagy writes in [25]:-

In his lecture at the 1928 International Congress of Mathematicians in Bologna, Riesz pointed out that the relevant parts of linear operations of continuous functions can be characterised and generated without using the Stieltjes integral. His method is based on the concept of the majorant operation. His basic theorem is that any majorant set of linear operations always has a smallest majorant. His method has the great advantage that it is completely general, so that it can be applied not only to linear operations of continuous functions, but also to functions interpreted on arbitrary abstract sets.

Of course although Riesz was happy in Szeged he would have liked to be appointed to the University of Budapest. An opportunity arose in 1936 when József Suták, who had been appointed in preference to Riesz in 1911, had retired from the Department of Higher Geometry in the Pázmány Péter University of Budapest. The University did not advertise the position but rather Lipót Fejér was asked by the Faculty to recommend to them the most suitable candidate. Fejér listed 21 Hungarian mathematicians, some working in Hungary and some abroad, who would be eligible. In his report he wrote that Frigyes Riesz was in a class of his own:-

... even among the excellent ones, a mathematician stands out, who, if we look at the weight of his pure mathematical discoveries and the time-tested nature of these, as well as the multitude of all his merits, is currently unmatched by anyone else.

After writing seven pages praising Riesz's outstanding qualities Fejér proposed to the Faculty that Riesz be invited to the Department of Advanced Geometry without an application. The Faculty met on 26 November 1936 and rejected this proposal without debate with 11 votes to accept and 29 votes to reject. No reason was given for the rejection but it is hard to see that the reason could have been any other than Riesz's Jewish origin.

In March 1944 German troops entered Hungary. Riesz was invited to give two lectures at the University of Geneva in the spring of 1944 but was unable to attend because of the problems he now faced. We note that the text of the two lectures he could not deliver in person were published as Sur la théorie ergodique Ⓣ (1945). In Szeged, Riesz was required to wear a yellow star, was forced to retire and was told he would be confined to a ghetto. The Swedish Embassy pleaded that he be permitted special treatment and he was allowed to stay in his apartment with the approval of the Minister of the Interior. He sewed on the yellow star, but always wore an overcoat over it. He retired his professorship on 1 August 1944. Despite the humiliation he was made to suffer, he survived. His brother Sándor Riesz also survived but, as we noted above, his sister Margit was murdered in Auschwitz together with her husband and daughter.

Géza Lakatos was appointed Prime Minister of Hungary in August 1944 in an effort to distance Hungary from Nazi Germany and seek a separate peace with the Allies. The new government rapidly reversed the decisions made on the Jews and Riesz was appointed rector of the University of Szeged in the second semester of the 1944-1945 academic year. In late 1945 Riesz was appointed to the chair of mathematics in the University of Budapest. His brother Marcel Riesz, who was living in Sweden, organised a three-month lecture tour of universities in Sweden and Denmark for Frigyes Riesz and from December 1947 to March 1948 he undertook this lecture tour.

Many of Riesz's fundamental findings in functional analysis were incorporated with those of Banach. He made many contributions to other areas including ergodic theory where he gave an elementary proof of the mean ergodic theorem in 1938. He also studied orthonormal series and topology. Rogosinski, writes of Riesz's style [22]:-

The work of F Riesz is not only distinguished by the genuine importance of his results, but also by his aesthetic discernment in mathematical taste and diction. ... The more leisurely mastership of F Riesz's style, whether he writes in his native Hungarian, or in French or German, conveys such pleasure and is to the older mathematician a nostalgic remainder of what we are in danger to lose. For him there was no mere abstraction for the sake of a structure theory, and he was always turning back to the applications in some concrete and substantial situation.

His book Leçon's d'analyse fonctionnelle Ⓣ (1952) is one of the most readable accounts of functional analysis ever written. Rogosinski describes this book, which Riesz wrote jointly with his student Béla Szökefalvi-Nagy, as follows [22]:-

Here, in the first half written by himself, we find the old master picturing to us Real Analysis as he saw it, lovingly, leisurely, and with the discerning eye of an artist. This book, I have no doubt, will remain a classic in the treasure house of mathematical literature. With it, and with all his other work, will live the memory of Frederic Riesz as a great and fertile mathematician for long in the history of our art.

For details of this book, including extracts from the Preface and from reviews, see THIS LINK.

For details about Frigyes Riesz by those who knew him, see THIS LINK.

Riesz received many honours for his work. In 1917 he received the Tomori Anasztáz Foundation Prize from the Hungarian Academy of Sciences. He had been elected to the Hungarian Academy of Sciences as a corresponding member in 1915 and was elected a full member in 1936. Among those who recommended him for full membership were Lipót Fejér and Béla Kerékjártó and they made clear that in the twenty years since he was made a corresponding member:-

... his earlier discoveries have had such a profound and far-reaching impact, and during this time he has paved the way for so many new research directions that have already sparked the most lively scientific movement, and that Frigyes Riesz is now recognised by mathematicians all over the world as a leading mathematician of the first order. ... The Riesz-Fischer theorem had already had numerous notable applications, but now it has been shown that in theoretical physics Heisenberg quantum mechanics is equivalent to Schrödinger quantum mechanics, and the old Riesz theorem provides the basis for proving this very important fact. ... As a researcher, teacher, and journal editor, Frigyes Riesz devotes all his strength to the service of his country, of which he is proud to be one of its foremost scientists.

The Hungarian Academy of Sciences awarded him its Marczibányi Prize in 1927, and its Grand Prize in 1946. In 1949 he was awarded its Gold Grade Kossuth Prize and in 1953 he was awarded its Kossuth Grand Prize. He received the Order of Merit of the People's Republic in 1950. He was elected a corresponding member of the Paris Academy of Sciences in 1948 and elected an external member of the Bavarian Academy of Sciences in 1954. He was also elected to the Royal Physiographic Society of Lund in Sweden. He received honorary doctorates from the universities of Szeged in 1946, Budapest in 1950 and Paris Sorbonne in 1954.

Riesz's functional analysis book was his last publication for his health began to fail. He spent his last six months in the Kútvölgyi sanatorium where he died in February 1956. After a small family funeral in the Reformed Church he was buried in the Kerepesi Cemetery (whose official name is the Fiume Road National Graveyard). We note that Loránd Eötvös and Lipót Fejér are also buried in Kerepesi Cemetery.