# Erhard Schmidt

### Quick Info

Born
13 January 1876
Dorpat, Russian Empire (now Tartu, Estonia)
Died
16 December 1959
Berlin, Germany

Summary
Schmidt is best remembered for the Gram-Schmidt orthogonalisation process, which takes a basis of a space and constructs an orthogonal one from it.

### Biography

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.

### References (show)

1. M Bernkopf, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. H Baumgärtel, Erhard Schmidt, John von Neumann, in Mathematics in Berlin (Berlin, 1998), 97-104.
3. A Dinghas, Erhard Schmidt (Erinnerungen und Werk), Jahresberichte der Deutschen Mathematiker-Vereinigung 72 (1) (1970/71), 3-17.
4. E Hellinger and O Toeplitz, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, Encyklopädie der Mathematischen Wissenschaften 13 (Leipzig, 1923-27), 1335-1602.
5. J Naas, Zum hundertsten Geburtstag von Erhard Schmidt (13.1.1876- 6.12.1959), Math. Nachr. 75 (1976), 7-8.
6. R Nevanlinna, Erhard Schmidt, zu seinem 80. Geburtstag, Math. Nachr. 15 (1956), 3-6.
7. H Rohrbach, Erhard Schmidt: Ein Lebensbild, Jahresberichte der Deutschen Mathematiker-Vereinigung 69 (4) (1) (1967/1968), 209-224.
8. K Schröder, Erhard Schmidt, Math. Nachr. 25 (1963), 1-3.
9. H Tietze, Erhard Schmidt: 13.1.1876- 6.12.1959, Bayer. Akad. Wiss. Jbuch. 1960 (1960), 176-177.