Introduction* The lattice of all closed subspaces of a separable Hubert space has the following properties. It is complete, atomic, irreducible, semi-modular, and orthocomplemented. The primary purpose of this paper is to investigate lattices with these properties. If L is such a lattice, there is a representation theorem for L. The elements in L of finite dimension or finite deficiency form an orthocomplemented modular lattice. It follows that if the dimension of L is high enough, then there

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... a dual pair of vector spaces U and W such that L is isomorphic to the lattice of W closed subspaces of U. Because L is orthocomplemented the spaces U and W are isomorphic. This isomorphism establishes a "semi-inner product" on U, and L may be described as being the lattice of closed subspaces of a semi-inner product space. The contents of the paper are as follows. Section 1 contains some definitions and establishes notation. Section 2 is concerned with the completion of an orthocomplemented lattice and § 3 with the center of such a lattice. With the exception of Theorem 3.2 the techniques used in § §2 and 3 are standard, and many of the results are widely known. To the best of the author's knowledge, however, the theorems have not previously appeared in print. Therefore we state and prove them in some detail. The representation theorem and other results centering about the semi-modularity condition are proved in §4. With the other conditions holding for L, semi-modularity is equivalent tσ certain covering conditions. Because this is not true for arbitrary complete atomic lattices, the results seem to be of some interest* Finally, in § 5, semi-inner product spaces are discussed. A theorem is given relating the existence of a semi-inner product on U to the existence of an orthocomplemented lattice of subspaces of U. This is an easy generalization of a theorem of Birkhoff and von Neumann [4] (Appendix). In two other theorems we investigate the exact relation between the semi-inner product on U and the orthocomplemented lattice L.