Erratum to: "infinitesimals in ordered vector spaces" Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2013, Том 15, Выпуск 2, С. 83-84

ЗАМЕТКИ

ERRATUM TO: "INFINITESIMALS IN ORDERED VECTOR SPACES"

E. Yu. Emelyanov

In this note, Theorem 1 in the article which is cited in the title is corrected.

We give the following Theorem 1 instead of Theorem 1 on page 21 in [1]. Theorem 1. Let V be an ordered space. Consider the following conditions:

(1) V is almost Archimedean;

(2) A(*V) П V = {0};

(3) A(*V) С n(*V);

(4) A(*V) С o-pns(*V);

(5) V is Archimedean.

Then (1) ^ (2) and (3) ^ (4) ^ (5).

< (1) ^ (2): It follows from the definition of A(*V).

(3) ^ (4): Just, since n(*V) С o-pns(*V).

(4) ^ (5): It is enough to show

V Э y < П u G V+ (Vn G N \{0})

^ [y < 0].

Take a y e V, such that y ^ nu e V+ for all n e N \ {0}. Fix some v e * N \ N. Then 1 u e A(*V) C o-pns(*V). Given z e V then, by the transfer principle, 1 u ^ z iff nu ^ z for some n e N \ {0}. Therefore,

u(- V) = U Un, where U.n = uf1 uj.

V n€N\{0} '

By the hypothe)is, infV (U( 1 u) - L( 1 u)) = 0. Hence infV U( 1 u) = infV (U( 1 u) - 0) = 0, since 0 G L (1 u). Thus

inf U Un = 0. (*)

V

nSN\{0}

Since y ^ n u e Un for all n e N \ {0} then it follows from (*) that y ^ 0, what is required.

(5) ^ (3): Let k e A(*V). Then -n u < k < n u for some u e V+ and all n e N \ {0}. In order to show k e n(*V), it is sufficient to prove that infV U(k) = 0. Take a w e U(k). Then —nu ^ k ^ w for all n e N \ {0}. Since V is Archimedean, infneN\{0} n u = 0 and

0 = — inf 1 u = sup | — 1 u) ^ w.

nSN\{0} n raeN\{0} V n '

© 2013 Emelyanov E. Yu.

84 Emelyanov E. Yu.

We obtain 0 ^ w, and hence 0 ^ U(k). Let V 3 z ^ U(k). Then z ^ n u for all n G N \ {0}. As V is Archimedean, we get z ^ 0. Thus, infV U(k) = 0, what is required. >

References

1. Emel'yanov E. Yu. Infinitesimals in ordered vector spaces // Vladikavkaz. Mat. Zh.—2013.—Vol. 15, № 1.—P. 18-22.

ИСПРАВЛЕНИЯ К СТАТЬЕ: «БЕСКОНЕЧНО МАЛЫЕ В УПОРЯДОЧЕННЫХ ВЕКТОРНЫХ ПРОСТРАНСТВАХ»

Емельянов Э. Ю.

Исправления внесены в формулировку Теоремы 1 на с. 21 статьи автора с указанным в заголовке названием, опубликованной во Владикавказском математическом журнале. 2013. Том 15, выпуск 1. С. 18-22.