艾尔哈德·施密特Erhard Schmidt

艾尔哈德·施密特的父亲是一位名叫Alexander Schmidt的医学生物学家。施密特的大学经历遵循了当时德国常见的一种模式，即学生在课程推进过程中在几所不同的大学学习。他先就读于当地的多尔帕特大学，然后前往柏林，在那里师从赫尔曼·阿曼杜斯·施瓦茨。

他于1905年在哥廷根大学在大卫·希尔伯特的指导下获得博士学位。他的学位论文题为Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener Ⓣ（按给定方案展开任意函数），是一部关于积分方程的著作。这篇论文的主要思想出现在我们下面描述的施密特1907年的论文中。获得博士学位后，他前往波恩，并于1906年在那里获得教授资格论文（Habilitation）。离开波恩后，施密特先后在苏黎世、埃尔朗根和布雷斯劳任职，之后于1917年被任命为柏林大学教授。这一任命是为了填补赫尔曼·阿曼杜斯·施瓦茨退休后空出的讲席。

施密特在费迪南德·格奥尔格·弗罗贝尼乌斯去世后不久抵达柏林大学，费迪南德·格奥尔格·弗罗贝尼乌斯曾与赫尔曼·阿曼杜斯·施瓦茨共同领导该系。另一位正教授是弗里德里希·赫曼·肖特基。康斯坦丁·卡拉西奥多里于1918年被任命，以填补费迪南德·格奥尔格·弗罗贝尼乌斯的讲席，并与施密特共同领导柏林的数学。然而，康斯坦丁·卡拉西奥多里在柏林只待了一年就离开了。施密特现在负有填补空缺讲席的主要责任。这被证明是一项困难的任务。施密特拟定了一份令人印象深刻的候选人名单：依次为勒伊岑·布劳威尔、赫尔曼·外尔和古斯塔夫·赫格洛茨。教授职位依次提供给这三人，但每人都拒绝了。接下来被提供该讲席的人是埃里希·赫克，他也拒绝了。该职位直到1921年才被填补，当时路德维希·比贝尔巴赫被提供该职位并接受了。同年，弗里德里希·赫曼·肖特基退休，而已经是柏林编外教授的伊赛·舒尔被提升为正教授。

我们讨论的这些职位都属于纯数学方面。当 施密特 到达柏林时，那里没有应用数学，这门学科被认为更适合技术学院。然而，施密特 是推动在柏林成立应用数学研究所的主要人物。研究所成立后，施密特 必须填补新设立的应用数学讲席和應用數學研究所所长的职位。他在1920年成功促成了一次极佳的任命，当时 von 理查德·冯·米泽斯 接受了这两个职位。亚历山大·奥斯特洛夫斯基 在1965年写道：-

只有随着 理查德·冯·米泽斯 被任命到柏林大学，第一个具有广泛影响力的、在数学上严肃的德国应用数学学派才得以存在。

把柏林带到应用数学这一领先地位，主要应归功于 施密特。显然，他的能力在数学之外也得到了认可，因为他被任命为1921-22学年的院长，并在1929-30年间担任柏林大学副校长。尽管这个职位具有全校性质，但他希望继续推动数学的愿望，从他就任副校长时所作的就职演说中可以看出：演说的题目是 On certainty in mathematics。

20世纪30年代对 施密特 来说是艰难的岁月。随着纳粹在1933年上台，施密特 的犹太同事的生活变得越来越困难，伊赛·舒尔、理查德·冯·米泽斯 和其他几个人被迫离职。1951年，在柏林举行了一次会议，庆祝 施密特 的75岁生日。汉斯·弗洛伊登萨 本人是一名在纳粹时期幸存下来的犹太人，他谈到了 施密特 在20世纪30年代经历的困难（例如见 [2]）：-

在数学本身中践行数学所要求的诚实是如此容易。如果你不这样做，你会很快受到严厉的惩罚。而要在面对人和朋友时坚持这种用数字和图形证明的美德，则要困难得多。我们这些身处外部、多年来被敌对的德国排斥的人知道这一点，并且从未对你们有过怀疑，这一点从大量来自国外的稿件到达⟦E1⟧编辑手中就可以看出。

在回复汉斯·弗洛伊登萨的致辞时，施密特谈到了他对柏林大学的热爱（例如见[2]）：-

我 simply 爱我的学生。对整所大学也完全一样。我爱柏林大学，无论它是否处于愉快的境况——这不会改变任何事情。从我来到柏林的时候起我就爱它，我将继续忠于它。

1936年，当问题非常困难时，施密特被任命为在奥斯陆举行的国际数学家大会德国代表团团长。在纳粹统治的这些艰难岁月里，施密特在柏林大学担任权威职位。他不得不执行针对犹太人的决议，但路德维希·比贝尔巴赫的一位助手在1938年报告说：-

我认为施密特完全不了解犹太人问题。

第二次世界大战结束后，II Schmidt被任命为德国科学院数学研究所所长。他一直担任该职务到1958年。那时他已经从讲席上退休，这是他在1950年做的，并且他已经不再担任数学系联合主任，这是1952年发生的。战后他还承担的另一个角色是Mathematische Nachrichten的第一任编辑。他在1948年参与创办了该期刊。

施密特的主要兴趣是积分方程和希尔伯特空间。他采纳了大卫·希尔伯特关于积分方程的各种想法，并在1905年左右将这些想法结合成大卫·希尔伯特空间的概念。大卫·希尔伯特在1904年研究了具有对称核的积分方程。他证明了在这种情况下积分方程有实特征值，这是大卫·希尔伯特的用词，而对应于这些特征值的解他称为特征函数。他还将核函数积分相关的函数展开为一组正交归一特征函数的无穷级数。

施密特在1907年发表了一篇关于积分方程的两部分论文，其中他以更简单的方式重新证明了大卫·希尔伯特的结果，而且限制更少。在这篇论文中，他给出了现在称为约尔根·佩尔森·格拉姆-施密特正交归一化过程的方法，用于从线性无关集构造正交归一函数集。然后他继续考虑核不对称的情况，并表明在这种情况下，与给定特征值相关的特征函数以伴随对的形式出现。

然而，我们应该注意到，皮埃尔·西蒙·拉普拉斯在约尔根·佩尔森·格拉姆或施密特之前就提出了约尔根·佩尔森·格拉姆-施密特过程。

1908年，施密特发表了一篇关于无穷多个未知数的无穷多个方程的重要论文，引入了各种几何符号和术语，这些符号和术语至今仍用于描述函数空间和内积空间。施密特的想法导致了希尔伯特空间的几何学，他当然必须被视为现代抽象泛函分析的创始人。

施密特定义了一个空间$H$，其元素是复数的平方可和序列。如果$w = \{w_{n}\}$和$z = \{z_{n}\}$是$H$的两个元素，施密特通过下式定义了一个内积

$(w,z) = \sum w_{n}z_{n}$。

他将元素$z$的范数||$z$||定义为$z$与其复共轭的内积的平方根。他定义了正交元素，表明由两两正交的元素组成的集合是线性无关的。他再次在此背景下给出了约尔根·佩尔森·格拉姆-施密特正交化过程。他还研究了投影和谱分解。今天所谓的大卫·希尔伯特-施密特算子也出现在这篇1908年的论文中。Bernkopf在[1]中写道：-

施密特在希尔伯特空间上的工作代表了向现代数学迈出的一大步。他是最早证明欧几里得概念的日常经验可以有意义地扩展到几何之外，进入更复杂的抽象数学的理想化构造的数学家之一。

施密特搬到柏林后，他的兴趣转向了拓扑学。他找到了卡米耶·若尔当曲线定理的一个新证明，这个证明很快成为经典。施密特对拓扑学的兴趣影响了海因茨·霍普夫，并且在1929年，他是海因茨·霍普夫博士论文的审查人。后来，施密特对等周不等式产生了兴趣，并于1949年发表了一篇关于此主题的重要论文。

Erhard Schmidt's father was a medical biologist called Alexander Schmidt. Erhard's university career followed a pattern which was common in Germany at this time, namely that students studied at several different universities as their course progressed. He attended his local university in Dorpat before going to Berlin where he studied with Schwarz.

His doctorate was obtained from the University of Göttingen in 1905 under Hilbert's supervision. His doctoral dissertation was entitled Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener Ⓣ and was a work on integral equations. The main ideas of this thesis appeared in Schmidt's 1907 paper which we describe below. After obtaining his doctorate he went to Bonn where he was awarded his habilitation in 1906. After leaving Bonn, Schmidt held positions in Zürich, Erlangen and Breslau before he was appointed to a professorship at the University of Berlin in 1917. The appointment was to fill the chair left vacant by Schwarz's retirement.

Schmidt arrived at the University of Berlin shortly after the death of Frobenius, who had jointly led the department with Schwarz. The other full professor was Schottky. Carathéodory was appointed in 1918 to fill Frobenius's chair and to jointly head mathematics in Berlin with Schmidt. However Carathéodory was to spend only one year in Berlin before leaving. Schmidt now had the main responsibility for filling the vacant chair. This proved a difficult task. Schmidt drew up an impressive list of candidates: Brouwer, Weyl, and Herglotz in that order. The professorship was offered to each of these in turn, with each turning it down. The next person to be offered the chair was Hecke who also turned it down. The position was not filled until 1921 when Bieberbach was offered the post and accepted it. In this same year Schottky retired and Schur, who was already an extraordinary professor in Berlin, was promoted to full professor.

The appointments we have discussed were on the pure mathematics side. When Schmidt arrived in Berlin there was no applied mathematics there, the subject being considered more suitable for technical colleges. However Schmidt was the main person who pushed for the founding of an Institute of Applied Mathematics in Berlin. After the Institute was set up Schmidt had to fill the new chair of applied mathematics and the post of Director of the Institute of Applied Mathematics. He was able to engineer a superb appointment in 1920 when von Mises accepted the two positions. Ostrowski wrote in 1965:-

Only with the appointment of Richard von Mises to the University of Berlin did the first mathematically serious German school of applied mathematics with a broad sphere of influence come into existence.

Credit for bringing Berlin to this leading role in applied mathematics must chiefly go to Schmidt. Clearly his abilities were recognised outside mathematics for he was appointed Dean for the academic year 1921-22 and the vice-chancellor of the University of Berlin during the years 1929-30. Despite the university wide nature of this post his wish to continue to promote mathematics is seen from the inaugural address he gave when taking up the post of vice-chancellor: it was entitled On certainty in mathematics.

The 1930s were difficult years for Schmidt. With the Nazi rise to power in 1933 life became increasingly difficult for Schmidt's Jewish colleagues and Schur, von Mises and several others were forced out of their posts. In 1951 a meeting was held in Berlin to celebrate Schmidt's 75th birthday. Hans Freudenthal, himself a Jew who had survived the Nazi years, spoke of Schmidt's difficulties through the 1930s (see for example [2]):-

It is so easy to practise the honesty that mathematics demands in mathematics itself. If you don't, you will be punished quickly and bitterly. It is so much more difficult to stick to this virtue, proven with numbers and figures, against humans and friends. That we outside, excluded for years from a hostile Germany, know this, and never doubted on you, this is evident from the large number of contributions from abroad that have reached the editors of the Festschrift.

In his reply to Freudenthal's address Schmidt spoke of his love of the University of Berlin (see for example [2]):-

I simply loved my students. And exactly the same is true of the university as a whole. I love the University of Berlin, whether it happens to be in happy conditions or not - this does not change anything. I have loved it from the time I have been in Berlin and I will remain faithful to it.

In 1936, when the problems were very difficult, Schmidt was made head of the German delegation to the International Congress of Mathematicians at Oslo. Schmidt held positions of authority at the University of Berlin through these difficult years of Nazi rule. He had to carry through the resolutions against Jews but one of Bieberbach's assistants reported in 1938:-

I think that Schmidt does not at all understand the Jewish question.

After the end of World War II Schmidt was appointed as Director of the Mathematics Research Institute of the German Academy of Science. He remained in that role until 1958. By that time he had retired from his chair, which he did in 1950, and he has ceased as joint head of the mathematics department, which happened in 1952. Another role which he took on after the end of the war was as the first editor of Mathematische Nachrichten. He had co-founded the journal in 1948.

Schmidt's main interest was in integral equations and Hilbert space. He took various ideas of Hilbert on integral equations and combined these into the concept of a Hilbert space around 1905. Hilbert had studied integral equations with symmetric kernel in 1904. He showed that in this case the integral equation had real eigenvalues, Hilbert's word, and the solutions corresponding to these eigenvalues he called eigenfunctions. He also expanded functions related to the integral of the kernel function as an infinite series in a set of orthonormal eigenfunctions.

Schmidt published a two part paper on integral equations in 1907 in which he reproved Hilbert's results in a simpler fashion, and also with less restrictions. In this paper he gave what is now called the Gram-Schmidt orthonormalisation process for constructing an orthonormal set of functions from a linearly independent set. He then went on to consider the case where the kernel is not symmetric and showed that in that case the eigenfunctions associated with a given eigenvalue occurred in adjoint pairs.

We should note, however, that Laplace presented the Gram-Schmidt process before either Gram or Schmidt.

In 1908 Schmidt published an important paper on infinitely many equations in infinitely many unknowns, introducing various geometric notations and terms which are still in use for describing spaces of functions and also in inner product spaces. Schmidt's ideas were to lead to the geometry of Hilbert spaces and he must certainly be considered as a founder of modern abstract functional analysis.

Schmidt defined a space $H$ whose elements are square summable sequences of complex numbers. If $w = \{w_{n}\}$ and $z = \{z_{n}\}$ are two elements of $H$, Schmidt defined an inner product by

$(w,z) = \sum w_{n}z_{n}$.

He defined the norm ||$z$|| of the element $z$ to be the square root of the inner product of $z$ with its complex conjugate. He defined orthogonal elements showing that a set consisting of pair-wise orthogonal elements was linearly independent. Again he gave the Gram-Schmidt orthonormalisation process in this setting. He also studied projections and spectral resolutions. What are today called Hilbert-Schmidt operators also appear in this 1908 paper. Bernkopf writes in [1]:-

Schmidt's work on Hilbert spaces represents a long step toward modern mathematics. He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics.

After Schmidt moved to Berlin his interests turned towards topology. He found a new proof of the Jordan curve theorem which quickly became a classic. Schmidt's interest in topology influenced Hopf and, in 1929, he was an examiner of Hopf's doctoral thesis. Later still Schmidt became interested in isoperimetric inequalities, publishing an important paper on this topic in 1949.