# On a Semigroup Problem II

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Dimensional Topological Vector Spaces

**Theorem**

**1**

**Theorem**

**2**

**.**Let X be a TVS of finite dimension d, not necessarily Hausdorff. Then, X is isomorphic to the Cartesian product ${\mathbb{R}}^{m}\times V$, where $m\le d$, and V is a topologically trivial TVS of dimension $d-m$.

**Theorem**

**3.**

**Proof.**

## 3. Infinite Dimensional Topological Vector Spaces

**Definition**

**1.**

**Remark**

**1.**

#### 3.1. The Semigroup S

#### 3.2. Normed Spaces

**Theorem**

**4.**

**Theorem**

**5**

**.**Assume X is an infinite-dimensional normed vector space. Then, there is a semigroup $S\subset X$ for which the semigroup problem fails. That is, S is not one side of any hyperplane, but its closure is not a group.

**Proof**

**of Theorem 5.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 3.3. Quasi-Normed Spaces

**Theorem 6.**

- X contains no basic sequence (see Definition 1)
- Every closed subspace of X with a separating dual is finite-dimensional.

**Definition**

**2.**

**Theorem**

**7**

**.**Every A-convex quasi-normed space (equivalently, any plurisubharmonic quasi-normed space) contains basic sequences.

**Theorem**

**8**

**.**Let X be a complete plurisubharmonic quasi-normed infinite-dimensional space. Then, there is a semigroup $S\subset X$ such that the closed linear span L of S has a separating dual, S is separated by each continuous non-zero functional on L, and the closure of S does not contain zero, so it is not a group.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Melbourne, I.; Niţică, V.; Tòrók, A. Stable transitivity of certain noncompact extensions of hyperbolic systems. Ann. Henri Poincaré
**2005**, 6, 725–746. [Google Scholar] [CrossRef] [Green Version] - Melbourne, I.; Niţică, V.; Tòrók, A. A note about stable transitivity of noncompact extensions of hyperbolic systems. Discrete Contin. Dyn. Syst.
**2006**, 14, 355–363. [Google Scholar] [CrossRef] - Niţică, V.; Tòrók, A. Stable transitivity of Heisenberg group extensions of hyperbolic systems. Nonlinearity
**2014**, 27, 661–683. [Google Scholar] [CrossRef] - Lui, K.; Niţică, V.; Venkatesh, S. The semigroup problem for central semidirect products of R
^{n}with R^{m}. Topol. Proc.**2015**, 45, 9–29. [Google Scholar] - Katok, A.; Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems; Encyclopedia of Mathematics and Its Applications; With a Supplementary Chapter by Katok and Leonardo Mendoza; Cambridge University Press: Cambridge, UK, 1995; Volume 54, p. xviii+802. [Google Scholar]
- Niţică, V.; Tòrók, A. Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review. Axioms
**2015**, 4, 84–101. [Google Scholar] [CrossRef] [Green Version] - Niţică, V.; Tòrók, A. On a semigroup problem. Discret. Contin. Dyn. Syst. Ser. S
**2019**, 12, 2365–2377. [Google Scholar] [CrossRef] [Green Version] - Silverman, T.; Miller, S.M. Hilbert Extensions of Anosov Diffeomorpisms. 2013. Available online: http://www.math.psu.edu/mass/reu/2013/mathfest/HilbertCounterexample.pdf (accessed on 5 August 2020).
- Rudin, W. Functional Analysis, 2nd ed.; International Series in Pure and Applied Mathematics; McGraw-Hill, Inc.: New York, NY, USA, 1991; p. xviii+424. [Google Scholar]
- Kelley, J.L.; Namioka, I. Linear Topological Spaces; Donoghue, W.F., Jr., Lucas, K.R., Pettis, B.J., Poulsen, E.T., Price, G.B., Robertson, W., Scott, W.R., Smith, K.T., Eds.; The University Series in Higher Mathematics; D. Van Nostrand Co., Inc.: Princeton, NJ, USA, 1963; p. xv+256. [Google Scholar]
- Diestel, J. Sequences and Series in Banach Spaces; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1984; Volume 92, p. xii+261. [Google Scholar] [CrossRef]
- Kalton, N.J. The basic sequence problem. Studia Math.
**1995**, 116, 167–187. [Google Scholar] [CrossRef] [Green Version] - Day, M.M. On the basis problem in normed spaces. Proc. Am. Math. Soc.
**1962**, 13, 655–658. [Google Scholar] [CrossRef] - Kalton, N.J.; Peck, N.T.; Roberts, J.W. An F—Space Sampler; London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK, 1984; Volume 89, p. xii+240. [Google Scholar] [CrossRef]
- Rolewicz, S. Metric Linear Spaces, 2nd ed.; PWN—Polish Scientific Publishers: Warsaw, Poland; D. Reidel Publishing Co.: Dordrecht, The Netherlands, 1984; p. xi+459. [Google Scholar]
- Etter, D.O., Jr. Vector-valued analytic functions. Trans. Am. Math. Soc.
**1965**, 119, 352–366. [Google Scholar] [CrossRef] - Kalton, N.J. Basic sequences in F-spaces and their applications. Proc. Edinb. Math. Soc.
**1974**, 19, 151–167. [Google Scholar] [CrossRef] [Green Version] - Kalton, N.J. Plurisubharmonic functions on quasi-Banach spaces. Studia Math.
**1986**, 84, 297–324. [Google Scholar] [CrossRef] - Tam, S.C. The basic sequence problem for quasi-normed spaces. Arch. Math.
**1994**, 62, 69–72. [Google Scholar] [CrossRef] - Niţică, V.; Pollicott, M. Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Theory Dynam. Syst.
**2005**, 25, 257–269. [Google Scholar] [CrossRef] - Aleksandrov, A.B. Essays on nonlocally convex Hardy classes. In Complex Analysis and Spectral Theory (Leningrad, 1979/1980); Lecture Notes in Math; Springer: Berlin, Germany; New York, NY, USA, 1981; Volume 864, pp. 1–89. [Google Scholar]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nitica, V.; Torok, A.
On a Semigroup Problem II. *Symmetry* **2020**, *12*, 1392.
https://doi.org/10.3390/sym12091392

**AMA Style**

Nitica V, Torok A.
On a Semigroup Problem II. *Symmetry*. 2020; 12(9):1392.
https://doi.org/10.3390/sym12091392

**Chicago/Turabian Style**

Nitica, Viorel, and Andrew Torok.
2020. "On a Semigroup Problem II" *Symmetry* 12, no. 9: 1392.
https://doi.org/10.3390/sym12091392