Lattice-Subspaces of C[0,1] and Positive Bases

AbstractA closed vector subspace of a vector lattice E is a lattice-subspace of E if X with the induced ordering is a vector lattice in its own right-but not necessarily a vector sublattice. In this work, we remark that C[0, 1] is a universal Banach lattice in the sense that every separable Banach lattice is order-isomorphic to a closed lattice-subspace of C[0, 1]. In addition, we show that (up to an order-isomorphism) if a closed lattice-subspace of C[0, 1] has a positive basis (bn), then for each n there exists a subinterval Jn on which bn is positive and every other element of this basis vanishes. By means of our main result, we present necessary and sufficient conditions that guarantee the existence of positive bases in an arbitrary lattice-subspace of C[0, 1]. These results are related to the general existence problem of positive bases in Banach lattices as well as the existence of unconditional bases in Banach spaces