Absorbing sets in a functional space related to Hausdorff dimension (in Ukrainian)

N.Mazurenko

It is proved that, for any sequence $(\gamma_i)$, $n < \gamma_1 < \gamma_2 < \dots < n + 1$, the sequence of the sets of functions in $C(\mathbb{I}^n)$ whose graphs are of Hausdorff dimension $> \gamma_i$ forms an $\mathcal{F}_\sigma$-absorbing sequence in $C(\mathbb{I}^n)$.

1. Van Mill J. The Infinite-Dimensional Topology of Function Spaces . -- Amsterdam: Elsevier, 2001. -- V. 64.-- 630~p.

2. Gladdines H. Absorbing systems in infinite-dimensional manifolds and applications . -- Amsterdam: Vrije Universiteit, 1994. -- 117~p.

3. Cauty R. Suites $\cal F_\sigma$-absorbantes en theorie de la dimension // Fundamenta Mathematicae.-- 1999. -- V. 159, no. 2. -- P.115--126.

4. Cauty R. Strong Universality and Its Applications // Proceedings of the Steklov Institute of Mathematics.~-- 1996. -- V.212. -- P.89--114.

5. Banakh T., Radul T., Zarichnyi M. Absorbing Sets in Infinite-Dimensional Manifolds. -- Lviv: VNTL Publishers, 1996. -- V. 1. -- 231~p.

6. Geoghegan R. On spaces of homeomorphisms, embeddings, and functions, II: The piecewise linear case // Proc. London Math. Soc. -- 1973. -- V. 3. -- P.463--483.

7. Dijkstra Jan J., Mogilski J. The topological product structure of systems of Lebesgue spaces // Math. Ann. -- 1991. -- V. 290. -- P.527--543.

8. Falconer K. J. The Geometry of Fractal Sets. -- Cambridge University Press, 1985. -- 162~p.

9. Wegmann H. Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen// J.~reine angew. Math. -- 1969. -- B. 234. -- 163--171.

10. Рудин У. Функциональный анализ. -- М.: Мир, 1976. -- 443~с.

11. Колмогоров А. Н., Фомин С. В. Элементы теории функций и функционального анализа. -- М.: Наука, 1976. -- 542~с.

12. Bessaga C., Pelczynski A. Selected topics in infinite-dimensional topology . -- Warszawa, 2001. -- 352~p.

13. Mazurenko N. Absorbing sets related to Hausdorff dimension // Visnyk Lviv Univ., Ser. Mech-Math. -- 2003. -- V.61. -- P.121--128.

	Absorbing sets in a functional space related to Hausdorff dimension (in Ukrainian)
Author	N.Mazurenko Vasyl Stefanyk Precarpathian National University
Abstract	It is proved that, for any sequence $(\gamma_i)$, $n < \gamma_1 < \gamma_2 < \dots < n + 1$, the sequence of the sets of functions in $C(\mathbb{I}^n)$ whose graphs are of Hausdorff dimension $> \gamma_i$ forms an $\mathcal{F}_\sigma$-absorbing sequence in $C(\mathbb{I}^n)$.
Keywords	absorbing set, functional space, Hausdorff dimension
DOI	doi:10.30970/ms.23.2.207-216
Reference	1. Van Mill J. The Infinite-Dimensional Topology of Function Spaces . -- Amsterdam: Elsevier, 2001. -- V. 64.-- 630~p. 2. Gladdines H. Absorbing systems in infinite-dimensional manifolds and applications . -- Amsterdam: Vrije Universiteit, 1994. -- 117~p. 3. Cauty R. Suites $\cal F_\sigma$-absorbantes en theorie de la dimension // Fundamenta Mathematicae.-- 1999. -- V. 159, no. 2. -- P.115--126. 4. Cauty R. Strong Universality and Its Applications // Proceedings of the Steklov Institute of Mathematics.~-- 1996. -- V.212. -- P.89--114. 5. Banakh T., Radul T., Zarichnyi M. Absorbing Sets in Infinite-Dimensional Manifolds. -- Lviv: VNTL Publishers, 1996. -- V. 1. -- 231~p. 6. Geoghegan R. On spaces of homeomorphisms, embeddings, and functions, II: The piecewise linear case // Proc. London Math. Soc. -- 1973. -- V. 3. -- P.463--483. 7. Dijkstra Jan J., Mogilski J. The topological product structure of systems of Lebesgue spaces // Math. Ann. -- 1991. -- V. 290. -- P.527--543. 8. Falconer K. J. The Geometry of Fractal Sets. -- Cambridge University Press, 1985. -- 162~p. 9. Wegmann H. Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen// J.~reine angew. Math. -- 1969. -- B. 234. -- 163--171. 10. Рудин У. Функциональный анализ. -- М.: Мир, 1976. -- 443~с. 11. Колмогоров А. Н., Фомин С. В. Элементы теории функций и функционального анализа. -- М.: Наука, 1976. -- 542~с. 12. Bessaga C., Pelczynski A. Selected topics in infinite-dimensional topology . -- Warszawa, 2001. -- 352~p. 13. Mazurenko N. Absorbing sets related to Hausdorff dimension // Visnyk Lviv Univ., Ser. Mech-Math. -- 2003. -- V.61. -- P.121--128.
Pages	207-216
Volume	23
Issue	2
Year	2005
Journal	Matematychni Studii
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