Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-05T21:22:24.525Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete Semi-Ordered Linear Spaces

Published online by Cambridge University Press:  20 November 2018

Israel Halperin  and
Hidegoro Nakano
Israel Halperin
Affiliation:
Queen's University
Hidegoro Nakano
Affiliation:
Tokyo 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.

Let R be a semi-ordered linear space, that is, a vector lattice in Birkhoff's terminology [2]. An element a ∈ R is said to be discrete, if for every element x ∈ R such that there exists a real number a for which x = aa. For every pair of discrete elements a, b ∈ R we have or there exists a real number a for which b = aa or a = ab.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 293 - 298
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Adams, C. R. and Morse, A. P., Random sampling in the evaluation of a Lebesgue integral, Bull. Amer. Math. Soc, vol. 45 (1939), 442–447.Google Scholar
[2] Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Pub. 25 (1940).Google Scholar
[3] Kantorovitch, L., Lineare halbgeordnete Raurne, Math. Sbornik, vol. 2 (44) (1937), 121–168.Google Scholar
[4] Nakano, H., Teilweise geordnete Algebra, Jap. J. Math., vol. 17 (1941), 425–511.Google Scholar
[5] Nakano, H., Stetige lineare Funktionale auf dem teilweisegeordneten Modul, J. Fac. Sci. Imp.Univ. Tokyo, vol. 4 (1942), 201–382.Google Scholar
[6] Nakano, H., Discrete semi-ordered linear spaces (in Japanese), Functional Analysis, vol. 1 (1947-9), 204–207.Google Scholar
[7] Nakano, H., Ergodic theorems in semi-ordered linear spaces, Ann. Math., vol. 49 (1948), 538–556.Google Scholar
[8] Nakano, H., Modulared semi-ordered linear spaces, Tokyo Math. Book Series, vol. 1 (Tokyo, 1950), §26, §27.Google Scholar
[9] Schur, J., Über lineare Transformationen in der Théorie der unendlichen Reihen, J. f. reine u. angew. Math., vol. 151 (1921), 79–111.Google Scholar