information:classification
Topological groups

Using his previous work on measure theory, von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups. He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers.

In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups. The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are Lie groups. This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.