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Lattice-subspaces of \(C[0,1]\) and positive bases. (English) Zbl 0802.46035

Summary: A 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 \((b_ n)\), then for each \(n\) there exists a subinterval \(J_ n\) on which \(b_ n\) 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.

MSC:

46E05 Lattices of continuous, differentiable or analytic functions
46B42 Banach lattices
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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