Differential Equations and Implicit Functions in Infinitely Many Variables.

will be discussed. The results of all three sections of the paper include as special cases the corresponding theorems on functions of a finite number of variables. The notion of a function / of infinitely many variables ( xy, x2, • • • ) was brought into mathematics in the wake of the infinite determinant, the consideration of infinite systems of linear equations and the researches of Hilbert, Hellinger, Toeplitz, Westfall, and others on bilinear and linear forms in infinitely many variables in connection with the study of integral equations. In the present paper, the region of points £ = (xy, x2, • • • ), in which the functions considered will be supposed defined, will be similar to the generalized parallelepipedon (3) R : \xí Oi\ eu (i = l,2, •••).