Hilbert's Fifth Problem (Q2736235)

In Problem V Hilbert asked whether in the definition of a Lie group the assumption of differentiability is necessary or does it follow from the remaining properties. The problem has attracted much attention and strongly influenced the development of the theory of Lie groups. The paper recalls the problem and the most general answer to it provided by \textit{I. Kaplansky} in 1971 [Lie algebras and locally compact groups, Chicago Univ. Press (1971; Zbl 0223.17001)]. NEWLINENEWLINENEWLINEThe second part of Problem V, related to functional equations, is not considered here [cf. \textit{J. Aczél}, Bull. Am. Math. Soc., New Ser. 20, 153-163 (1989; Zbl 0676.39004)].NEWLINENEWLINEFor the entire collection see [Zbl 0902.00028].