Bendixson was born August 1, 1861 in Djurgårdsbrunn, StockholmSweden to a middle-class family. His father Vilhelm Emanuel Bendixson was a merchant, and his mother was Tony Amelia Warburg. On completing secondary education in Stockholm, he obtained his school certificate on May 25, 1878. On September 13, 1878 he enrolled to the Royal Institute of Technology in Stockholm. In 1879 Bendixson went to Uppsala University and graduated with the equivalent of a Master's degree on January 27, 1881. Graduating from Uppsala, he went on to study at the newly opened Stockholm University College after which he was awarded a doctorate by Uppsala University on May 29, 1890. On June 10, 1890 Bendixson was appointed as a docent at Stockholm University College. He then worked as an assistant to the professor of mathematical analysis from March 5, 1891 until May 31, 1892. From 1892 until 1899 he taught at the Royal Institute of Technology and he also taught calculus and algebra at Stockholm University College. During this period he married Anna Helena Lind on December 19, 1887. Anna, who was about eighteen months older than Bendixson, was the daughter of the banker Johan Lind. In 1899 Bendixson substituted for the Professor of Pure Mathematics at the Royal Institute of Technology and then he was promoted to professor there on January 26, 1900. On June 16, 1905 he became professor of higher mathematical analysis at Stockholm University College and from 1911 until 1927 he was its rector. For his outstanding contributions, Bendixson received many honours including an honorary doctorate on May 24, 1907. Bendixson became more involved in politics as his career progressed. He was well known for his mild left-wing views and he put his beliefs into practice being head of a committee to help poor students. He served on many other committees and he was an advisor to a committee which investigated a proportional representationvoting system in Sweden in 1912–13. In this capacity he was able to make use of his mathematical skills in advising the committee.
Scientific achievements
Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems from applied mathematics. His first research work was on set theory and the foundations of mathematics, following the ideas which Georg Cantor had introduced. He contributed important results in point set topology. As a young student Bendixson made his name by proving that every uncountableclosed set can be partitioned into a perfect set and a countable set. He also gave another important contribution when he gave an example of a perfect set which is totally disconnected. Concerning solution of a polynomial equation by radicals Bendixson returned to Niels Henrik Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals. The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points. The Poincaré–Bendixson theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Henri Poincaré but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901. In 1902, he derived Bendixson's inequality which puts bounds on the characteristic roots of matrices.
