Vector spaces with additive topology and distinctly continuous multiplication by a scalar

Maslyuchenko Volodymyr Kyrylovych

Maslyuchenko Volodymyr Kyrylovych ¹

¹ Department of Mathematical Analysis, Yuriy Fedkovych Chernivtsi National University, Chernivtsi, 58000, Ukraine

Keywords: additive topology, a scalar, Mazur-Orlicz theorem

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Abstract

The notion of $N$ -space is introduced, is $N$ -space is a vector space with a topology, in which the addition operation is jointly continuous and multiplication on scalar is separately continuous. The notion o f bounded set is also introduced in such a space. We obtain the following generalization of Mazur-Orlicz theorem about joint continuity of scalar multiplication in $N$ spaces: If $X$ is $N$-space in which every convergent sequence is bounded, then scalar multiplication in this space is jointly sequentially continuous

References

[1] Banakh S.S. Course of functional analysis. - K.: Soviet school, 1948. - 216 p.

[2] Mazur S., Orlicz W. Über Folgen linearer Operationen / / Stud. Math. 1933. - 4. - S. 152-157.

[3] Hellinger E., Teeplitz O . Grunndlagen für eine Theorie der unendlichen Matrizen / / Math. Ann. - 1910. 69 . - S. 289-330.

[4] Bourbaki H. Topological vector spaces. - M.: IL, 1959. - 410 p.

[5] Edwards P. Functional analysis. Theory and applications. - M.: Mir, 1969. - 1072 p.

[6] Rudin W. Functional Analysis. - M.: Mir, 1975. - 445 p.

[7] Masliuchenko V.K.. Algebraic aspects of the science of relations between the cut and aggregate properties of functions // All-Ukrainian scientific conference “Modern problems of probability theory and mathematical analysis”. Abstracts. February 23-28, 2011, Vorokhta. - Ivano-Frankivsk: PNU, 2011. - P.2-4.

[8] Maslyuchenko V.K. The first types of topological vector spaces. - Chernivtsi: Ruta, 2002. - 72 p.

[9] Maslyuchenko V.K.. Linear continuous operators. - Chernivtsi: Ruta, 2002. - 72 p.

[10] Maslyuchenko V.K.. Elements of the theory of duality. - Chernivtsi: Ruta, 2005. - 160 p.

[11] Maslyuchenko V.K.. Lectures on functional analysis. Ч .1 . Metric and normalized spaces. - Chernivtsi: CHNU, 2010. - 184 p.

[12] Maslyuchenko V.K.. Lectures on functional analysis. Ч .2 . Linear operators and functionalities. - Chernivtsi: CHNU, 2010. - 192 p.

[13] Maslyuchenko V.K. Vector spaces with additive topology and separately continuous product // "Complex analysis and its appl." Intern. Conf.ded. 70-th ann. A.F.Grishin. B ook o f Abstract. Kharkiv, August 15-18, 2011. - Kharkiv: KhNU, 2011. - P.31.

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Cite

ACS Style: Maslyuchenko, V.K. Vector spaces with additive topology and distinctly continuous multiplication by a scalar. Bukovinian Mathematical Journal. 2018, 1
AMA Style: Maslyuchenko VK. Vector spaces with additive topology and distinctly continuous multiplication by a scalar. Bukovinian Mathematical Journal. 2018; 1(4).
Chicago/Turabian Style: Volodymyr Kyrylovych Maslyuchenko. 2018. "Vector spaces with additive topology and distinctly continuous multiplication by a scalar". Bukovinian Mathematical Journal. 1 no. 4.

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