Hostname: page-component-7bb8b95d7b-fmk2r Total loading time: 0 Render date: 2024-10-02T14:21:41.653Z Has data issue: false hasContentIssue false
  • English
  • Français

Separation and Approximation in Topological Vector Lattices

Published online by Cambridge University Press:  20 November 2018

Solomon Leader
Solomon Leader*
Affiliation:
Rutgers University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.

The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 286 - 296
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Birkhoff, G., Lattice theory, A.M.S. Coll. Pub. (New York, 1940).Google Scholar
2. Birkhoff, G. and Pierce, R. S., Lattice-ordered rings, Anais da Acad. Brasileira de Ciencias, 28 (1956), 41–69.Google Scholar
3. Bochner, S., Additive set functions on groups, Ann. Math., 40 (1939), 769–99.Google Scholar
4. Bochner, S. and Phillips, R. S., Additive set functions and vector lattices, Ann. Math., 4 (1941), 316-24.Google Scholar
5. Freudenthal, H., Teilweise geordnete Moduln, Proc. Acad. Wet. Amsterdam, 89 (1936), 641–51.Google Scholar
6. Leader, S., The theory of Lp-spaces for finitely additive set functions, Ann. Math., 58 (1953), 528–43.Google Scholar
7. Lebesgue, H., Leçons sur Vintégration et la recherche des fonctions primitives, Gauthier- Villars (Paris, 1928).Google Scholar
8. Moore, E. H. and Smith, H. L., A general theory of limits, Amer. J. Math., 44 (1922), 102-21.Google Scholar
9. Nakano, H., Modern spectral theory (Tokyo, 1950).Google Scholar
10. Namioka, I., Partially ordered linear topological spaces, Amer. Math. Soc, Mem. 24 (1957).Google Scholar
11. Saks, S., Theory of the integral (Warsaw, 1937).Google Scholar
12. Stone, M. H., Applications of the theory of boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.Google Scholar