adj. 1. (of an element of an ordering or lattice) having no element greater than it, being the greatest element of a chain. A maximal element need not be the unique greatest element unless the ordering is total; for example, the set of proper subsets of a given set ordered by inclusion has no greatest element, but every set formed by removing a single member from the given set is maximal. See maximum. Compare minimal. 2. (of an orthogonal or orthonormal sequence) such that whenever the inner products of any given element with the sequence elements are zero, so is the given element. Unless the inner product space is complete, not every maximal orthonormal sequence need be a Schauder basis, but clearly every basis is maximal.