From 1911 until 1920, Fischer was professor at the University of Erlangen, appointed to fill the chair left vacant in the previous year when Paul Gordan retired. Emmy Noether had been awarded her doctorate from the University of Erlangen in 1907 having worked under Gordan's supervision. When Fischer arrived in Erlangen it was natural for Noether to work with him. After Noether's death in 1935, Weyl gave an address at which he spoke of Fischer's influence:-
Fischer's field was algebra ..., in particular the theory of elimination and of invariants. He exerted upon Emmy Noether, I believe, a more penetrating influence than Gordan did. Under his direction the transition from Gordan's formal standpoint to the Hilbert method of approach was accomplished. She refers in her papers at this time again and again to conversations with Fischer.Fischer took part in World War I from 1915 to 1918. He married Ellis Strauss, the daughter of Pfarrers Eugen Strauss, in Erlangen in 1917. Fischer was 42 years old, his wife being 26; they had one daughter. From 1920 Fischer worked at the University of Cologne, remaining there until he retired in 1938.
Fischer is best known for one of the highpoints of the theory of Lebesgue integration, called the Riesz-Fischer Theorem. The theorem is that the space of all square-integrable functions is complete, in the sense that Hilbert space is complete, and the two spaces are isomorphic by means of a mapping based on a complete orthonormal system.
Let us note again the major result, the Riesz-Fischer Theorem, for which he is best known as Weyl noted in the above quote. In 1907 Ernst Fischer studied orthonormal sequences of functions and gave necessary and sufficient conditions for a sequence of constants to be the Fourier coefficients of a square integrable function. His two papers of 1907 were Sur la convergence en moyenne Ⓣ and Applications d'un théorèm sur la convergence en moyenne Ⓣ both published in Comptes rendus of the Academy of Sciences in Paris. This work led to the concept of a Hilbert space. Frigyes Riesz published a similar result in the same year. The theorem, now called the Riesz-Fischer theorem, is one of the great achievements of the Lebesgue theory of integration.
Fischer went on to study Hadamard determinants, publishing his results in 1908 in the Archiv der Mathematik und Physik, and Sylvester determinants, publishing a paper in Crelle's Journal in the following year. He also published in the Carathéodory Problem and on finite abelian groups.
Article by: J J O'Connor and E F Robertson
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