scholarly journals On a Semigroup Problem II

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1392
Author(s):  
Viorel Nitica ◽  
Andrew Torok
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinite Dimensional ◽  
Finite Dimensional

We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative.

EMBEDDINGS OF FREE TOPOLOGICAL VECTOR SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 311-324
Author(s):  
ARKADY LEIDERMAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Dimensional Subspace ◽  
Vector Subspace ◽  
Compact Metric Space ◽  
Topological Vector Spaces ◽  
Finite Dimensional ◽  
Bernstein Set ◽  
Dimensional Cube ◽  
Open Question

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

scholarly journals On properly separable quotients of strict (LF) Spaces

1989 ◽  
Vol 47 (2) ◽  
pp. 307-312 ◽  
Author(s):  
W. J. Robertson
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Inductive Limit ◽  
General Question ◽  
Fréchet Spaces ◽  
Infinite Dimensional

AbstractAll known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

scholarly journals Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Liaqat Ali Khan ◽  
Saud M. Alsulami
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Almost Periodic Functions ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Range Space ◽  
Finite Dimensional ◽  
Locally Convex ◽  
Dimensional Range

The notion of asymptotic almost periodicity was…first introduced by Fréchet in 1941 in the case of…finite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity.

scholarly journals On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mohamed Benslimane ◽  
Abdelhadi Moutassim
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Alternative Algebra ◽  
Jordan Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
First Case ◽  
Hilbert Norm ◽  
Complex Powers

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

scholarly journals Linearization of certain uniform homeomorphisms

2007 ◽  
Vol 82 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Anthony Weston
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Infinite Dimensional

AbstractThis article concerns the uniform classification of infinite dimensional real topological vector spaces. We examine a recently isolated linearization procedure for uniform homeomorphisms of the form φ: X →Y, where X is a Banach space with non-trivial type and Y is any topological vector space. For such a uniform homeomorphism φ, we show that Y must be normable and have the same supremal type as X. That Y is normable generalizes theorems of Bessaga and Enflo. This aspect of the theory determines new examples of uniform non-equivalence. That supremal type is a uniform invariant for Banach spaces is essentially due to Ribe. Our linearization approach gives an interesting new proof of Ribe's result.

Decidable subspaces and recursively enumerable subspaces

1984 ◽  
Vol 49 (4) ◽  
pp. 1137-1145 ◽  
Author(s):  
C. J. Ash ◽  
R. G. Downey
Keyword(s):  
Vector Space ◽  
Order Theory ◽  
Infinite Dimensional ◽  
First Order ◽  
Direct Sum ◽  
First Order Theory ◽  
Finite Dimensional ◽  
Recursively Enumerable ◽  
Natural Analogues

AbstractA subspace V of an infinite dimensional fully effective vector space V∞ is called decidable if V is r.e. and there exists an r.e. W such that V ⊕ W = V∞. These subspaces of V∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V∞) and the set of decidable subspaces forms a lower semilattice S(V∞). We analyse S(V∞) and its relationship with L(V∞). We show:Proposition. Let U, V, W ∈ L(V∞) where U is infinite dimensional andU ⊕ V = W. Then there exists a decidable subspace D such that U ⊕ D = W.Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces.These results allow us to show:Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V∞), is undecidable.This contrasts sharply with the result for recursive sets.Finally we examine various generalizations of our results. In particular we analyse S*(V∞), that is, S(V∞) modulo finite dimensional subspaces. We show S*(V∞) is not a lattice.

Probability Measures with Big Kernels

2001 ◽  
Vol 8 (2) ◽  
pp. 333-346
Author(s):  
Nguyen Duy Tien ◽  
V. Tarieladze
Keyword(s):  
Probability Measure ◽  
Vector Space ◽  
Topological Vector Space ◽  
Dual Space ◽  
Full Measure ◽  
Probability Measures ◽  
Infinite Dimensional ◽  
Cylindrical Measure ◽  
Second Category

Abstract It is shown that in an infinite-dimensional dually separated second category topological vector space X there does not exist a probability measure μ for which the kernel coincides with X. Moreover, we show that in “good” cases the kernel has the full measure if and only if it is finitedimensional. Also, the problem posed by S. Chevet [Kernel associated with a cylindrical measure, Springer-Verlag, 1981, p. 69] is solved by proving that the annihilator of the kernel of a measure μ coincides with the annihilator of μ if and only if the topology of μ-convergence in the dual space is essentially dually separated.

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