The free abelian topological group as a subgroup of the free locally convex topological vector space

2003 ◽  
Vol 6 (3) ◽  
Author(s):  
Carolyn E. McPhail
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Locally Convex

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

scholarly journals A class of factorable topological algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 53-59 ◽  
Author(s):  
E. Ansari-Piri
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Approximate Identity ◽  
Necessary Condition ◽  
Topological Algebras ◽  
Approximate Identities ◽  
Cohen Factorization ◽  
Locally Convex ◽  
Space Property ◽  
Uniformly Bounded

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

scholarly journals Asymptotic Farkas lemmas for convex systems

2016 ◽  
Vol 19 (4) ◽  
pp. 160-168
Author(s):  
Dinh Nguyen ◽  
Mo Hong Tran
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Constraint Qualification ◽  
Convex Set ◽  
Lower Semicontinuous ◽  
Closed Convex Cone ◽  
Convex Mapping ◽  
Hausdorff Topological Vector Space ◽  
Reverse Convex ◽  
Locally Convex

In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

Mathematika ◽  
2019 ◽  
Vol 65 (3) ◽  
pp. 708-718 ◽  
Author(s):  
Mikołaj Krupski ◽  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Topological Group ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group

scholarly journals Finitely spectral operators

1986 ◽  
Vol 28 (1) ◽  
pp. 95-112 ◽  
Author(s):  
B. Nagy
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Topological Vector Space ◽  
General Situation ◽  
Countable Additivity ◽  
Resolution Of The Identity ◽  
Set Function ◽  
Locally Convex

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.

scholarly journals Mean value theorems and a Taylor theorem for vector valued functions

1981 ◽  
Vol 24 (1) ◽  
pp. 69-77
Author(s):  
Rudolf Výborný
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Extension Theorem ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

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