44. A Screw Line in Hilbert Space and its Application to the Probability Theory

1. A. Kolmogoroff has investigated %he speeralzation of the screw line in Hilbert space in his paper "Kuven in Hilbertschen Raum, die gegeniber einer einparametrigen Gruppen yon Bewegung invarian sind ", where he has promised to give the complete proofs in another paper. In this note i will show his results, although the proofs may run in the same way as his own. And I will apply the results to the theory of two-dimensional brownian motions. 2. Under a congruent transformation in a Hilbert space we understand an isometric mapping from (C) to (C) itself. On account of the Mazur-Ulam’s theorem any congruent transformation K is expressible in the form"