A characterization of projective spaces by a set of planes

This note deals with the following question: How many planes of a linear space (P, L) must be known as projective planes to ensure that (P, L) is a projective space? The following answer is given: If for any subset M of a linear space (P, L) the restriction (M, L)(M)) is locally complete, and if for every plane E of (M, L(M)) the plane (E) over bar generated by E is a projective plane, then (P, L) is a projective space. Or more generally: If for any subset M of P the restriction (M, L(M)) is locally complete, and if for any two distinct coplanar lines G(1), G(2) is an element of L(M) the lines (G) over bar(1), (G) over bar(2) is an element of L generated by G(1), G(2) have a nonempty intersection and <(G(1) boolean OR G(2))over bar> satisfies the exchange condition, then (P, L) is a generalized projective space.