Existence and convergence of probability measures in Banach spaces.

Theorems of the Bochner-Sazonov type are proved for Banach spaces with a basis. These theorems give sufficient conditions of a topological nature under which a positive definite function is the characteristic functional of a probability measure. The conditions are, in a certain natural sense, best possible. Central limit theorems of the Lindeberg type for triangular systems of random variables taking values in a Banach space with a basis are obtained. Applications to l p {l_p} and C [ 0 , 1 ] C[0,1] are given.