On the algebraic dimension of Riesz spaces

  • N. M. Baziv Vasyl Stefanyk Precarpathian National University
  • O. B. Hrybel Vasyl Stefanyk Precarpathian National University

Анотація

We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.

Біографії авторів

N. M. Baziv, Vasyl Stefanyk Precarpathian National University

Vasyl Stefanyk Precarpathian National University

O. B. Hrybel, Vasyl Stefanyk Precarpathian National University

Vasyl Stefanyk Precarpathian National University

Посилання

C.D. Aliprantis, O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006.

T. Banakh, A. Plichko, The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fнs. Nat. Ser. A Mat., 100 (2006), №1-2, 31–37.

T. Banakh, A. Plichko, Letter to editors from T. Banakh, A. Plichko, RACSAM. Rev. R. Acad. Cienc. Exactas Fнs. Nat. Ser. A Mat., 102 (2008), №2, 203.

C. Bessaga, A. Pe lczy´nski, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Warszawa, 1975.

J.R. Buddenhagen, Classroom notes: subsets of a countable set, Amer. Math. Monthly 78 (1971), №5, 536–537.

M.M. Day, Normed linear spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

H.E. Lacey, The Hamel dimension of any infinite dimensional separable Banach space is c,, Amer. Math. Monthly, 80 (1973), №3, 298.

W.A.J. Luxemburg, A.C. Zaanen, Riesz spaces. Vol. 1, North Holland Publ. Comp., Amsterdam–London, 1971.

V. Mykhaylyuk, M. Pliev, M. Popov, The lateral order on Riesz spaces and orthogonally additive operators, Positivity 25 (2021), №2, 291–327. DOI: 10.1007/s11117-020-00761-x.

V. Mykhaylyuk, M. Pliev, M. Popov, O. Sobchuk, Dividing measures and narrow operators, Studia Math., 231 (2015), №2, 97–116.

Опубліковано
2021-10-23
Як цитувати
Baziv, N. M., & Hrybel, O. B. (2021). On the algebraic dimension of Riesz spaces. Математичні студії, 56(1), 67-71. https://doi.org/10.30970/ms.56.1.67-71
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