Abstract
The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space Ω, unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice <L of the universally complete vector lattice C ∞ (Ω) of all extended-real-valued continuous functions f on Ω for which % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaacmqabaGaeqyYdCNaeyicI4SaeyyQdCLaeyOoaOJaaiiFaiab% gkzaMkabgIcaOiabgM8a3jabgMcaPiaacYhacqGH9aqpcqGHEisPai% aawUhacaGL9baaaaa!4E05!\[\left\{ {\omega \in \Omega :|f(\omega )| = \infty } \right\}\] is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that Ľ consists of bounded functions only.
The aim of this short article is to present a simple complete characterization of such vector lattices.
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References
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Abramovich, Y.A., Filter, W. A remark on the representation of vector lattices as spaces of continuous real-valued functions. Acta Appl Math 27, 23–26 (1992). https://doi.org/10.1007/BF00046632
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DOI: https://doi.org/10.1007/BF00046632