爱德华·海涅Eduard Heine

爱德华·海涅的父母是银行家KarlHeinrich Heine和Henriette Maertens。海涅是他父母九个孩子中的第八个。他在家里接受早期教育，然后进入Friedrichswerdersche文理中学（Gymnasium）。他后来转到柏林科隆文理中学学习，那里教授数学和自然科学。1838年秋天高中毕业考试（Abitur）后，海涅进入柏林大学。一个学期后，他去了哥廷根大学，在那里他听了卡尔·弗里德里希·高斯和M A莫里茨·亚伯拉罕·斯特恩关于数论的讲座。

在哥廷根三个学期后，海涅回到柏林大学，在那里由约翰·彼得·古斯塔夫·勒热纳·狄利克雷授课。他还听了雅各布·施泰纳的几何讲座和天文台台长约翰·弗朗茨·恩克的天文学讲座。然而，他也结识了柏林其他杰出的数学家：卡尔·魏尔斯特拉斯、恩斯特·爱德华·库默尔、利奥波德·克罗内克和卡尔·威廉·博尔夏特。他一生都与这些数学家保持联系，多次前往柏林。他留在柏林进行研究，由恩诺·德克森和马丁·欧姆指导。海涅提交了他的学位论文De aequationibus nonnullis differentialibusⓉ（一些微分方程），并于1842年4月30日获得学位。学位论文献给了约翰·彼得·古斯塔夫·勒热纳·狄利克雷。此后，他在柯尼斯堡花了一年时间跟随卡尔·古斯塔夫·雅各布·雅可比和恩斯特·弗朗茨·诺伊曼学习。他参加了柯尼斯堡数学讨论班，Aronhold、古斯塔夫·基尔霍夫和路德维希·赛德尔也参加了。

1844年7月20日，海涅在波恩大学habilitated，被任命为privatdozent。1848年5月13日，他在那里晋升为编外教授；1850年，他与柏林一位商人的女儿Sophie Wolff结婚；他们有五个孩子，四个女儿和一个儿子。1856年9月6日，海涅在哈勒晋升为正教授。在到达哈勒之前，海涅发表了关于偏微分方程的论文；在哈勒最初几年的教学中，他撰写了关于热理论、级数求和、连分数和椭圆函数的论文。

海涅教授过多种课程，如：位势论及其应用、数论、Fourier series、三角级数、力学和热理论。Wangerin于1862年在哈勒读本科，听过海涅的讲座，他写道：-

作为学术教师，海涅深受喜爱。他的讲座以清晰著称，广受好评。他总是向学生指出，仅仅阅读教科书和手册是不够的，他们有必要研究原始论文中的方法。

海涅研究了阿德里安-马里·勒让德多项式、加布里尔·拉梅函数和Bessel functions。他最为人所铭记的是海涅-埃米尔·博雷尔定理：-

……实数的一个子集是紧致的，当且仅当它是闭且有界的。

海涅还提出了一致连续的概念。

论文[2]分析了海涅写的、1988年在儒勒·昂利·庞加莱研究所发现的书信。该论文的第二部分涵盖了海涅-埃米尔·博雷尔定理的历史，并在以下评论中进行了总结：-

论文的后半部分致力于更系统地叙述所谓的海涅-Borel定理的逐步发现和表述。它从该定理在闭有界区间上连续函数一致连续这一定理的各种证明中的隐含使用开始。这一定理的第一个证明由约翰·彼得·古斯塔夫·勒热纳·狄利克雷在其1862年的讲座中给出（1904年出版），早于海涅在1872年证明它。Dugac表明，约翰·彼得·古斯塔夫·勒热纳·狄利克雷比海涅更明确地使用了覆盖和有限子覆盖的思想。这一思想也被卡尔·魏尔斯特拉斯和萨尔瓦托雷-平谢尔使用。埃米尔·博雷尔在1895年对可数覆盖表述了他的定理，而阿图尔·舍恩弗利斯和昂利·勒贝格分别在1900年和1898年（1904年出版）将其推广到任何类型的覆盖。Dugac表明，实际情况要复杂得多，还包括Cousin、卡尔·约翰尼斯·托马、Young、Vieillefond、恩斯特·列奥纳德·林德勒夫等名字。优先权问题通过引用昂利·勒贝格与埃米尔·博雷尔之间的通信和其他信件得到了很好的说明。

Eduard Heine's parents were Karl Heinrich Heine, a banker, and Henriette Maertens. Eduard was the eighth of his parents nine children. He received his early instruction at home, then entered the Friedrichswerdersche Gymnasium. He later changed to study at the Köllnische Gymnasium in Berlin where mathematics and natural sciences were taught. After his Abitur in the autumn 1838, Heine entered the University of Berlin. After one semester he went to the University of Göttingen where he attended lectures by Gauss and by M A Stern on number theory.

After three semesters at Göttingen, Heine returned to the University of Berlin where he was taught by Dirichlet. He also attended geometry lectures by Steiner, and astronomy lectures by J F Encke, the director of the observatory. However, he also got to know the other outstanding mathematicians in Berlin: Weierstrass, Kummer, Kronecker, and Borchardt. Throughout his life he maintained his contacts with these mathematicians, making numerous journeys to Berlin. He remained at Berlin to undertake research which was supervised by Enno Dirksen and Martin Ohm. Heine submitted his thesis De aequationibus nonnullis differentialibus Ⓣ and was awarded the degree on 30 April 1842. The thesis was dedicated to Dirichlet. Following this he spent a year at Königsberg studying with Jacobi and Franz Neumann. He took part in the Königsberg mathematical semiar which was also attended by Aronhold, Kirchhoff, and Seidel.

On 20 July 1844 Heine habilitated at the University of Bonn where he was appointed as a privatdozent. There he was promoted to extraordinary professor on 13 May 1848 and in 1850 he married Sophie Wolff, the daughter of a merchant from Berlin; they have five children, four daughters and a son. On 6 September 1856 Heine was promoted to ordinary professor at Halle. Before arriving at Halle, Heine published on partial differential equations and during his first few years teaching at Halle he wrote papers on the theory of heat, summation of series, continued fractions and elliptic functions.

At Halle, Heine taught a variety of courses such as: potential theory and its applications, number theory, Fourier series, trigonometric series, mechanics, and the theory of heat. Wangerin, who was an undergraduate in Halle in 1862, and attended lectures by Heine, wrote:-

As academic teacher Heine was greatly liked. His lectures were distinguished by clarity and were well received. He always pointed out to his students that it is not sufficient to read text books and manuals, rather it was necessary for them to study the approach in original papers.

Heine worked on Legendre polynomials, Lamé functions and Bessel functions. He is best remembered for the Heine-Borel theorem:-

... a subset of the reals is compact if and only if it is closed and bounded .

Heine also formulated the concept of uniform continuity.

The paper [2] analyses letters written by Heine and found in 1988 in the Institut Henri Poincaré. The second part of this paper covers the history of the Heine-Borel theorem and is summarised in the following review:-

The last half of the paper is devoted to a more systematic account of the gradual discovery and formulation of the so-called Heine-Borel theorem. It begins with the implicit use of the theorem in various proofs of the theorem stating that a continuous function on a closed, bounded interval is uniformly continuous. The first proof of this theorem was given by Dirichlet in his lectures of 1862 (published 1904) before Heine proved it in 1872. Dugac shows that Dirichlet used the idea of a covering and a finite subcovering more explicitly than Heine. This idea was also used by Weierstrass and Pincherle. Borel formulated his theorem for countable coverings in 1895 and Schönflies and Lebesgue generalized it to any type of covering in 1900 and 1898 (published 1904), respectively. Dugac shows that the story is in fact much more complicated and includes names such as Cousin, Thomae, Young, Vieillefond, Lindelöf. The priority questions are nicely illustrated with quotes from the correspondence between Lebesgue and Borel and other letters.