2024/04/03 by Omid Zabeti, Zabeti, Omid
Computer Science · Engineering · Mathematics · #Advanced Algebra and Logic #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Scheduling and Optimization Algorithms
paper · pdf · doi:10.48550/arxiv.2404.02851
openalex publication_date 2024/04/03 · openalex created_date 2024/04/06 · openalex updated_date 2026/08/03
For an arbitrary topological space X, assume that S(X) is the vector lattice of all equivalence classes of real-valued continuous functions on open dense subsets of X; it is a laterally complete vector lattice but not a normed lattice, certainly. Nevertheless, we can have the extended unbounded norm topology (un-topology) on it. On the other hand, by a remarkable result of Wickstead, there exists a representation approach for every Archimedean vector lattice E in terms of S(X)-spaces. In this paper, we show that this representation is order continuous and when E is order complete, it coincides with the known Maeda-Ogasawara representation. Moreover, when E is a Banach lattice, by consideration of the un-topology on E and the extended un-topology on S(X), we show that this representation is, in fact, a homeomorphism. With the aid of this topological attitude, we establish a representation theorem (in fact a homeomorphism) for the Fremlin projective tensor product between Banach lattices, in terms of S(X)-spaces, as well.