Frigyes Riesz studied at Budapest. He went to Göttingen and Zurich to further his studies, and obtained his doctorate from Budapest in 1902. His doctoral dissertation was on geometry. He spent two years teaching in schools before being appointed to a university post.

Riesz was a founder of functional analysis and his work has many important applications in physics. He built on ideas introduced by Fréchet in his dissertation, using his ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt.

In 1907 and 1909, Riesz produced representation theorems for a functional on quadratic Lebesgue integrable functions and in terms of a Stieltjes integral. The following year he introduced the space of q-fold Lebesgue integrable functions and so he began the study of normed function spaces. Riesz introduced the idea of the "weak convergence" of a sequence of functions. A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz.

Riesz's work of 1910 marks the start of operator theory. In 1918 his work came close to an axiomatic theory for Banach spaces, which were set up axiomatically 2 years later by Banach in his dissertation.

Riesz was appointed to a chair in Hungary in 1911. In 1922 Riesz set up the János Bolyai Mathematical Institute in a joint venture with Haar. Riesz became editor of the newly founded journal of the Institute Acta Scientiarum Mathematicarum which quickly became a major source of mathematics. Riesz was to publish many papers in this journal, the first in 1922 being on Egorov's theorem on linear functionals. It was published in the first part of the first volume. In 1945, Riesz was appointed to the chair of mathematics in the University of Budapest.

Many of Riesz's fundamental findings in functional analysis were incorporated with those of Banach. The theorem, now called the Riesz-Fischer theorem, which he proved in 1907 is fundamental in the Fourier analysis of Hilbert space. It was the mathematical basis for proving that matrix mechanics and wave mechanics were equivalent. This is of fundamental importance in early quantum theory. Riesz made many contributions to other areas including ergodic theory where he gave an elementary proof of the mean ergodic theorem in 1938. He also studied orthonormal series and topology. His book on functional analysis is one of the most readable accounts of the subject ever written.

Riesz received many honours for his work. He was elected to the Hungarian Academy of Science. Paris Academy of Sciences. and Royal Physiographic Society of Lund in Sweden. He received honorary doctorates from the universities of Szeged, Budapest and Paris.