In the paper there are considered random variables, the values of which are taken from a separable HILBERT space. By introducing a scalar product one may define the correlation coefficient between such random variables and basing on this definition the maximal correlation. Using a conception developped by CZAKI and FISCHER one gets the connections between the correlation coefficient, the canonical correlations, the vector correlation, the multiple correlation coefficient, the alienation coefficient, the maximal correlation. Further the author calculates the maximal correlation for random variables which are normally distributed in a separable HILBERT space. For the correlation coefficient and the canonical correlations he gives consistent estimations.