Linear spaces with many projective planes

Abstract.

We assume that in a linear space $(P, \frak{L})$ there is a non-empty set M of points with the property that every plane containing a point of M is a projective plane. In section 3 an example is given that in general $(P, \frak{L})$ is not a projective space. But if M can be completed by two points to a generating set of P, then $(P, \frak{L})$ is a projective space.