斯特凡·巴拿赫Stefan Banach

斯特凡·巴拿赫的父亲是约瑟夫·斯特凡 Greczek。首先要注意的是，巴拿赫不是他父亲的姓，但巴拿赫被赋予了他父亲的名字。斯特凡 Greczek是一名税务官员，他没有与巴拿赫的母亲结婚，后者在斯特凡受洗后便从场景中消失了，当时他只有四天大，关于她再无更多所知。在斯特凡的出生证明上，母亲一栏所写的名字是Katarzyna 巴拿赫。一些人认为她曾是斯特凡母亲的仆人，而另一些人则声称她是一名洗衣妇，在斯特凡很小的时候照顾过他。在晚年，巴拿赫试图查明他的母亲是谁，但他的父亲拒绝说任何话，只说他曾发誓对她的身份保密。

斯特凡 Greczek出生在一个叫Ostrowsko的小村庄，位于克拉科夫以南约50公里处。巴拿赫受洗后，正是被带到Ostrowsko，带到他祖母的家。然而，当巴拿赫的祖母生病时，斯特凡 Greczek安排他的儿子由住在克拉科夫的Franciszka Plowa和她的女儿Maria抚养。尽管巴拿赫从未回到祖母身边生活，但他在成长过程中确实经常去看望她。Maria的监护人是法国知识分子Juliusz Mien，他很快认出了巴拿赫所具有的才能。Mien教这个小男孩说法语，并且总体上让他懂得了教育的价值。

巴拿赫在克拉科夫上小学，1902年离开该校，开始在克拉科夫的Henryk Sienkiewicz文理中学（Gymnasium）第四中学接受中等教育。一个幸运的巧合是，巴拿赫班上的一个学生是Witold Wilkosz，他本人后来成为数学教授。这所学校似乎并不是特别好，1906年Wilkosz离开，转到一所更好的文理中学。然而，巴拿赫仍留在Henryk Sienkiewicz第四文理中学，尽管他与Wilkosz保持着联系。

在文理中学的最初几年，巴拿赫取得了头等成绩，数学和自然科学是他最好的科目。一位同学回忆巴拿赫人生中的这一时期（见[3]）：-

[巴拿赫]与同事相处愉快，但在数学之外对任何事物都不感兴趣。如果他开口说话，他会说得非常快，快得就像他进行数学思考一样。……Wilkosz也是类似的现象。他们两人之间，没有哪个数学问题不能迅速解决。此外，虽然巴拿赫在数学问题上更快，Wilkosz在解决物理问题方面快得惊人，而巴拿赫对此毫无兴趣。

他早年的优异成绩随着临近学校毕业考试而让位于较差的成绩。他在1910年通过了这次考试，但未能以优异成绩通过，这一荣誉大约授予四分之一的学生。离开学校后，巴拿赫和Wilkosz都想学习数学，但两人都觉得数学中不可能发现新东西，因此各自选择从事数学以外的学科。巴拿赫选择学习工程，Wilkosz选择学习东方语言。两位如此杰出的未来数学家竟会因为这个原因做出这样的决定，必定意味着没有人能适当地指导他们。

巴拿赫的父亲从未给过儿子多少支持，但如今他一离开学校，父亲就相当公开地告诉巴拿赫，他现在得靠自己了。巴拿赫离开克拉科夫前往伦贝格（今乌克兰利沃夫），在那里进入伦贝格技术大学工程学院就读。几乎可以肯定，巴拿赫没有任何经济支持，不得不靠做家教来养活自己。这必定占去了他相当多的时间，因此当他1914年毕业时，完成学业所用的时间比正常情况更长。1910年至1914年在伦贝格学习期间，他经常回到克拉科夫。巴拿赫在1914年的计划究竟是什么并不完全清楚，但在他毕业不久后的8月，第一次世界大战爆发，巴拿赫离开了伦贝格。

巴拿赫在那里学习时，伦贝格处于奥地利控制之下，自1772年瓜分波兰以来一直如此。在巴拿赫的青年时代，波兰在某种意义上并不存在，俄罗斯控制着该国大部分地区。华沙只有一所俄语大学，位于当时被称为“维斯瓦河地区”的地方。第一次世界大战爆发后，俄军占领了伦贝格城。巴拿赫因左眼视力不佳，身体条件不适合服兵役。战争期间，他从事筑路工作，但也在克拉科夫度过了一段时间，在当地学校教书挣钱。他还在克拉科夫雅盖隆大学听数学讲座，尽管这一点并不完全确定，但据信他听过斯塔尼斯瓦夫·萨伦巴的讲座。

1916年春天发生了一件偶然事件，对巴拿赫的生活产生了重大影响。一直在服兵役的胡戈·施泰因豪斯即将赴伦贝格的扬·卡齐米日大学任职。然而1916年春天他住在克拉科夫，等待就任。他傍晚会在克拉科夫的街道上散步，正如他在回忆录中所述：-

在一次这样的散步中，我无意中听到“昂利·勒贝格测度”这几个字。我走近公园长椅，向两位年轻的数学学徒做了自我介绍。他们告诉我，他们还有一位同伴，名叫Witold Wilkosz，他们对他赞不绝口。这两个年轻人是巴拿赫和欧顿·尼科迪姆。从那时起，我们定期见面，并且……我们决定建立一个数学学会。

这张长椅上巴拿赫和欧顿·尼科迪姆雕像的照片见THIS LINK。

胡戈·施泰因豪斯告诉巴拿赫一个他一直在研究却未能成功的问题。几天后，巴拿赫有了所需反例的主要想法，胡戈·施泰因豪斯和巴拿赫写了一篇联合论文，交给斯塔尼斯瓦夫·萨伦巴发表。战争延误了发表，但这篇论文——巴拿赫的第一篇论文——于1918年发表在Bulletin of the Kraków Academy上。从他与胡戈·施泰因豪斯做出这些最初成果之时起，巴拿赫开始以极快的速度产出重要的数学论文。当然，无法断言如果没有与胡戈·施泰因豪斯的偶然相遇，巴拿赫是否会走上数学研究的道路。也正是通过胡戈·施泰因豪斯，巴拿赫结识了他未来的妻子Lucja Braus。他们于1920年在山地度假胜地扎科帕内结婚。

在胡戈·施泰因豪斯的倡议下，Mathematical Society of Kraków于1919年成立。斯塔尼斯瓦夫·萨伦巴主持了成立大会，并当选为该学会首任主席。巴拿赫在1919年向该学会作了两次演讲，并继续产出高质量的研究论文。Mathematical Society of Kraków于1920年成为波兰数学会。

巴拿赫于1920年获得了在利沃夫技术大学（位于今乌克兰利沃夫）担任巴拿赫的助手职位。他在那里讲授数学，并在巴拿赫的指导下提交了博士学位论文。当然，这不是获得博士学位的标准途径，因为巴拿赫没有大学数学学历。然而，破例允许他提交On Operations on Abstract Sets and their Application to 积分方程。这篇学位论文[1]：

……有时被认为标志着泛函分析的诞生。

1922年，利沃夫的扬·卡齐米日大学授予巴拿赫教授资格论文（Habilitation）学位，其学位论文是关于测度论的。1921-22学年的大学日历报告[3]：

1922年4月7日，根据院务委员会决议，巴拿赫博士获得了数学讲师（Docent）的任教资格。根据1922年7月22日国家元首颁布的法令，他被任命为该学科的特别教授。

1924年，巴拿赫晋升为正教授，并在1924-25学年在巴黎度过。两次世界大战之间的岁月对巴拿赫来说极其忙碌。除了继续发表一系列重要论文外，他还为高中编写了算术、几何和代数教材。他还非常参与数学出版工作。1929年，他与胡戈·施泰因豪斯一起创办了一份新期刊Studia Mathematica，巴拿赫和胡戈·施泰因豪斯成为首批编辑。编辑方针是：

……专注于泛函分析及相关主题的研究。

另一项重要的出版事业始于1931年，是一套新的Mathematical Monographs丛书。这套丛书由来自Lwów的巴拿赫和胡戈·施泰因豪斯，以及来自华沙的Knaster、卡齐米日·库拉托夫斯基、斯特凡·马祖尔凯维奇和瓦茨瓦夫·谢尔宾斯基担任编辑。该丛书的第一卷Théorie des Opérations linéairesⓉ（线性运算理论）由巴拿赫撰写，于1932年出版。这是他1931年最初以波兰语出版的著作的法语版本，并迅速成为经典。1936年，巴拿赫在奥斯陆举行的国际数学家大会上作了全会报告。在这次报告中，他描述了整个Lwów学派的工作，并且还谈到了他们进一步发展其思想的计划。

对巴拿赫的另一重要影响是卡齐米日·库拉托夫斯基于1927年被任命到利沃夫技术大学，并在那里工作到1934年。巴拿赫与卡齐米日·库拉托夫斯基合作，在此期间他们合写了一些论文。

巴拿赫的工作方式是非传统的。他喜欢在Lwów的咖啡馆里与同事们一起做数学。斯坦尼斯瓦夫·乌拉姆在[4]中回忆了在苏格兰咖啡馆的频繁聚会：-

在这些聚会中，要在耐力或酒量上胜过巴拿赫是很困难的。我们讨论当场提出的问题，往往经过数小时的思考后仍无明显解答。第二天，巴拿赫很可能带着几张小纸片出现，上面写着他已完成的证明概要。

你可以在THIS LINK看到苏格兰咖啡馆的照片。

Andrzej Turowicz也是Lwów的Kazimierz大学的数学教授，他也描述了巴拿赫的工作风格（见[3]）：-

[巴拿赫]大部分时间都在咖啡馆度过，不仅是与他人为伴，也独自一人。他喜欢那里的噪音和音乐。这些并不妨碍他集中精力和思考。有时，咖啡馆夜间关门后，他会走到火车站，那里的自助餐厅通宵营业。在那里，他一边喝着一杯啤酒，一边思考他的问题。

1939年，就在第二次世界大战开始前，巴拿赫当选为波兰数学会的主席。战争初期，苏联军队占领了Lwów。巴拿赫在战争开始前就与苏联数学家关系良好，曾多次访问莫斯科，因此受到新苏联政府的优待。他获准继续担任大学的讲席，并成为该大学理学院的院长，该大学此时已更名为伊万·弗兰科大学。巴拿赫的父亲为躲避向克拉科夫推进的德军而来到Lwów。在这个阶段，巴拿赫的生活几乎没有变化，他继续进行研究、撰写教科书、讲课以及在咖啡馆聚会。舍盖·索伯列夫和帕维尔·亚历山德罗夫于1940年在Lwów拜访了巴拿赫，而巴拿赫则在苏联参加了一些会议。德国入侵苏联时，他正在基辅，随即返回Lwów的家中。

1941年6月纳粹占领利沃夫意味着巴拿赫生活在极其困难的条件下。他因涉嫌买卖德国货币而被捕，但几周后获释。他经历了一段波兰学者被杀害的时期，他的博士导师Lomnicki在1941年7月3日那个发生许多屠杀的悲惨夜晚死去。1941年底，巴拿赫在德国传染病研究所喂虱子。喂虱子成了他在纳粹占领利沃夫剩余时间直至1944年7月的生活。苏联军队一收复利沃夫，巴拿赫就重新建立了联系。他在莫斯科郊外会见了舍盖·索伯列夫，但显然此时他已病重。舍盖·索伯列夫在巴拿赫纪念会议上致辞时谈到这次会面（例如见[3]）：-

尽管德国占领下的战争岁月留下了沉重痕迹，尽管严重的疾病正在削弱他的体力，巴拿赫的眼睛仍然充满生气。他仍然是战前我在利沃夫见到的那个善于交际、愉快、极其善良而迷人的巴拿赫。他在我记忆中就是这样：极富幽默感，精力充沛，心灵美好，才华出众。

巴拿赫计划战后去克拉科夫担任雅盖隆大学数学讲席，但他于1945年在利沃夫死于肺癌。

巴拿赫创立了现代分析，并对topological向量空间理论作出了重大贡献。此外，他还对测度论、积分、集合论和正交级数作出了贡献。

在他1920年撰写的学位论文中，他公理化地定义了今天所谓的巴拿赫空间。大约同一时期，其他人也提出了这一思想，例如诺伯特·维纳引入了这个概念，但没有发展出理论。“巴拿赫空间”这个名称是莫里斯·弗雷歇创造的。巴拿赫代数也以他的名字命名。

巴拿赫空间是一个实或复赋范向量空间，它在由范数诱导的度量下作为度量空间是完备的

$d(x, y) = ||x-y||$

。完备性很重要，因为这意味着巴拿赫空间中的奥古斯丁·路易·柯西序列收敛。

巴拿赫代数是一个巴拿赫空间，其中范数满足

$||xy|| ≤ ||x||.||y||$

巴拿赫贡献的重要性在于他发展了系统的泛函分析理论，而在此之前只有一些孤立的结果，后来人们才发现这些结果可以纳入这一新理论。该理论推广了维多·沃尔泰拉、埃里克·伊瓦尔·弗雷德霍姆和大卫·希尔伯特在积分方程方面的贡献。

巴拿赫证明了若干关于赋范线性空间的基本结果，今天许多重要定理都以他的名字命名。有关于连续线性泛函延拓的汉斯·哈恩-巴拿赫定理，关于有界映射族的巴拿赫-胡戈·施泰因豪斯定理，巴拿赫-Alaoglu定理，巴拿赫不动点定理，以及球的巴拿赫-阿尔弗雷德·塔斯基悖论分解。

巴拿赫-阿尔弗雷德·塔斯基悖论出现在1926年两位数学家在Fundamenta Mathematicae上发表的联合论文中，题为Sur la décomposition des ensembles de points en parties respectivement congruent Ⓣ（论点集分解为分别全等的部分）。这个令人困惑的悖论表明，一个球可以分成若干子集，这些子集可以拼合起来构成两个球，每个都与第一个球完全相同。需要选择公理来定义这种分解，而它能够给出如此非直观的结果，这一事实使一些数学家质疑该公理的使用。巴拿赫-阿尔弗雷德·塔斯基悖论是对这一时期公理集合论研究工作的重大贡献。

巴拿赫1929年的开映射也使用了集合论概念，这次是勒内-路易·贝尔在其1899年学位论文中引入的概念。

Stefan Banach's father was Stefan Greczek. The first thing to notice is that Banach was not his father's surname, but Banach was given his father's first name. Stefan Greczek was a tax official who was not married to Banach's mother who vanished from the scene after Stefan was baptised, when he was only four days old, and nothing more is known of her. The name given as Stefan's mother on his birth certificate is Katarzyna Banach. She is thought by some to have been the servant of Stefan's mother, while others claim that she was a laundress who took care of Stefan when he was very young. In later life Banach tried to find out who his mother was but his father refused to say anything except that he had been sworn to secrecy over her identity.

Stefan Greczek was born in a small village called Ostrowsko, some 50 km south of Kraków. It was to Ostrowsko, to his grandmother's home, that Banach was taken after his baptism. However, when Banach's grandmother took ill, Stefan Greczek arranged for his son to be brought up by Franciszka Plowa who lived in Kraków with her daughter Maria. Although Banach never went back to live with his grandmother, he did visit her frequently as he grew up. Maria's guardian was a French intellectual Juliusz Mien and he quickly recognised the talents that Banach had. Mien taught the young boy to speak French and in general gave him an appreciation for education.

Banach attended primary school in Kraków, leaving the school in 1902 to begin his secondary education at the Henryk Sienkiewicz Gymnasium No 4 in Kraków. By a fortunate coincidence, one of the students in Banach's class was Witold Wilkosz who himself went on to become a professor of mathematics. The school does not appear to have been a particularly good one and in 1906 Wilkosz left to move to a better Gymnasium. Banach, however, remained at Henryk Sienkiewicz Gymnasium No 4 although he maintained contact with Wilkosz.

During his first few years at the Gymnasium Banach achieved first class grades with mathematics and natural sciences being his best subjects. A fellow school pupil recalled Banach at this period in his life (see [3]):-

[Banach] was pleasant in dealings with his colleagues, but outside of mathematics he was not interested in anything. If he spoke at all, he would speak very rapidly, as rapidly as he thought mathematically. ... Wilkosz was a similar phenomenon. Between the two of them there was no mathematical problem that they could not speedily tackle. Also, while Banach was faster in mathematical problems, Wilkosz was phenomenally fast in solving problems in physics, which were of no interest to Banach.

The excellent grades of his early years gave way to poorer grades as he approached his final school examination. He passed this examination in 1910 but he failed to achieve a pass with distinction, an honour which went to about one quarter of the students. On leaving school Banach and Wilkosz both wanted to study mathematics, but both felt that nothing new could be discovered in mathematics so each chose to work in a subject other than mathematics. Banach chose to study engineering, Wilkosz chose oriental languages. That two such outstanding future mathematicians could make a decision for this reason must mean that there was nobody to properly advise them.

Banach's father had never given his son much support, but now once he left school he quite openly told Banach that he was now on his own. Banach left Kraków and went to Lemberg (now Lviv in Ukraine) where he enrolled in the Faculty of Engineering at Lemberg Technical University. It is almost certain that Banach, without any financial support, had to support himself by tutoring. This must have occupied quite a lot of his time and when he graduated in 1914 he had taken longer to complete the course than was normal. He had returned to Kraków frequently during the period of his studies in Lemberg from 1910 to 1914. It is not entirely clear what Banach's plans were in 1914 but the outbreak of World War I in August, shortly after his graduation, saw Banach leave Lemberg.

Lemberg was, at the time Banach studied there, under Austrian control as it had been from the partition of Poland in 1772. In Banach's youth Poland, in some sense, did not exist and Russia controlled much of the country. Warsaw only had a Russian language university and was situated in what was named "Vistula Land". With the outbreak of World War I, the Russian troops occupied the city of Lemberg. Banach was not physically fit for army service, having poor vision in his left eye. During the war he worked building roads but also spent time in Kraków where he earned money by teaching in the local schools. He also attended mathematics lectures at the Jagiellonian University in Kraków and, although this is not completely certain, it is believed that he attended Zaremba's lectures.

A chance event occurred in the spring of 1916 which was to have a major impact on Banach's life. Steinhaus, who had been undertaking military service, was about to take up a post at the Jan Kazimierz University in Lemberg. However he was living in Kraków in the spring of 1916, waiting to take up the appointment. He would walk through the streets of Kraków in the evenings and, as he related in his memoirs:-

During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society.

A picture of statues of Banach and Nikodym on this bench is at THIS LINK.

Steinhaus told Banach of a problem which he was working on without success. After a few days Banach had the main idea for the required counterexample and Steinhaus and Banach wrote a joint paper, which they presented to Zaremba for publication. The war delayed publication but the paper, Banach's first, appeared in the Bulletin of the Kraków Academy in 1918. From the time that he produced these first results with Steinhaus, Banach started to produce important mathematics papers at a rapid rate. Of course it is impossible to say whether, without the chance meeting with Steinhaus, Banach would have followed the route of research in mathematics. It was also through Steinhaus that Banach met his future wife Lucja Braus. They were married in the mountain resort of Zakopane in 1920.

On Steinhaus's initiative, the Mathematical Society of Kraków was set up in 1919. Zaremba chaired the inaugural meeting and was elected as the first President of the Society. Banach lectured to the Society twice during 1919 and continued to produce top quality research papers. The Mathematical Society of Kraków went on to became the Polish Mathematical Society in 1920.

Banach was offered an assistantship to Lomnicki at Lwów Technical University (in what is now Lviv in Ukraine) in 1920. He lectured there in mathematics and submitted a dissertation for his doctorate under Lomnicki's supervision. This was, of course, not the standard route to a doctorate, for Banach had no university mathematics qualifications. However, an exception was made to allow him to submit On Operations on Abstract Sets and their Application to Integral Equations. This thesis [1]:-

... is sometimes said to mark the birth of functional analysis.

In 1922 the Jan Kazimierz University in Lwów awarded Banach his habilitation for a thesis on measure theory. The University Calendar for 1921-22 reports [3]:-

On 7 April 1922, by resolution of the Faculty Council, Dr Stefan Banach received his habilitation for a Docent of Mathematics degree. He was appointed Professor Extraordinary of that subject by decree of the Head of State issued on 22 July 1922.

In 1924 Banach was promoted to full professor and he spent the academic year 1924-25 in Paris. The years between the wars were extremely busy one for Banach. As well as continuing to produce a stream of important papers, he wrote arithmetic, geometry and algebra texts for high schools. He also was very much involved with the publication of mathematics. In 1929, together with Steinhaus, he started a new journal Studia Mathematica and Banach and Steinhaus became the first editors. The editorial policy was:-

... to focus on research in functional analysis and related topics.

Another important publishing venture, begun in 1931, was a new series of Mathematical Monographs. These were set up under the editorship of Banach and Steinhaus from Lwów and Knaster, Kuratowski, Mazurkiewicz, and Sierpiński from Warsaw. The first volume in the series Théorie des Opérations linéaires Ⓣ was written by Banach and appeared in 1932. It was a French version of a volume he originally published in Polish in 1931 and quickly became a classic. In 1936 Banach gave a plenary address at the International Congress of Mathematicians in Oslo. In this address he described the work of the whole of the Lwów school, and he also spoke of the plans which they had to develop their ideas further.

Another important influence on Banach was the fact that Kuratowski was appointed to the Lwów Technical University in 1927 and worked there until 1934. Banach collaborated with Kuratowski and they wrote some joint papers during this period.

The way that Banach worked was unconventional. He liked to do mathematical with his colleagues in the cafés of Lwów. Ulam recalls in [4] frequent sessions in the Scottish Café:-

It was difficult to outlast or outdrink Banach during these sessions. We discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed.

You can see a picture of the Scottish Café at THIS LINK.

Andrzej Turowicz, also a professor of mathematics at the Kazimierz University in Lwów, also described Banach's style of working (see [3]):-

[Banach] would spend most of his days in cafés, not only in the company of others but also by himself. He liked the noise and the music. They did not prevent him from concentrating and thinking. There were cases when, after the cafés closed for the night, he would walk over to the railway station where the cafeteria was open around the clock. There, over a glass of beer, he would think about his problems.

In 1939, just before the start of World War II, Banach was elected as President of the Polish Mathematical Society. At the beginning of the war Soviet troops occupied Lwów. Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration. He was allowed to continue to hold his chair at the university and he became the Dean of the Faculty of Science at the university, now renamed the Ivan Franko University. Banach's father came to Lwów fleeing from the German armies advancing towards Kraków. Life at this stage was little changed for Banach who continued his research, his textbook writing, lecturing and sessions in the cafés. Sobolev and Aleksandrov visited Banach in Lwów in 1940, while Banach attended conferences in the Soviet Union. He was in Kiev when Germany invaded the Soviet Union and he returned immediately to his family in Lwów.

The Nazi occupation of Lwów in June 1941 meant that Banach lived under very difficult conditions. He was arrested under suspicion of trafficking in German currency but released after a few weeks. He survived a period when Polish academics were murdered, his doctoral supervisor Lomnicki dying on the tragic night of 3 July 1941 when many massacres occurred. Towards the end of 1941 Banach worked feeding lice in German institute dealing with infectious diseases. Feeding lice was to be his life during the remainder of the Nazi occupation of Lwów up to July 1944. As soon as the Soviet troops retook Lwów Banach renewed his contacts. He met Sobolev outside Moscow but clearly he was by this time seriously ill. Sobolev, giving an address at a memorial conference for Banach, said of this meeting (see for example [3]):-

Despite heavy traces of the war years under German occupation, and despite the grave illness that was undercutting his strength, Banach's eyes were still lively. He remained the same sociable, cheerful, and extraordinarily well-meaning and charming Stefan Banach whom I had seen in Lwów before the war. That is how he remains in my memory: with a great sense of humour, an energetic human being, a beautiful soul, and a great talent.

Banach planned to go to Kraków after the war to take up the chair of mathematics at the Jagiellonian University but he died in Lwów in 1945 of lung cancer.

Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.

In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space. The idea was introduced by others at about the same time, for example Wiener introduced the notion but did not develop the theory. The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him.

A Banach space is a real or complex normed vector space that is complete as a metric space under the metric

$d(x, y) = ||x-y||$

induced by the norm. The completeness is important as this means that Cauchy sequences in Banach spaces converge.

A Banach algebra is a Banach space where the norm satisfies

$||xy|| ≤ ||x||.||y||$

The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory. The theory generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations.

Banach proved a number of fundamental results on normed linear spaces, and many important theorems are today named after him. There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.

The Banach-Tarski paradox appeared in a joint paper of the two mathematicians in 1926 in Fundamenta Mathematicae entitled Sur la décomposition des ensembles de points en parties respectivement congruent Ⓣ. The puzzling paradox shows that a ball can be divided up into subsets which can be fitted together to make two balls each identical to the first. The axiom of choice is needed to define the decomposition and the fact that it is able to give such a non-intuitive result has made some mathematicians question the use of the axiom. The Banach-Tarski paradox was a major contribution to the work being done on axiomatic set theory around this period.

Banach's open mapping of 1929 also uses set-theoretic concepts, this time concepts introduced by Baire in his 1899 dissertation.