Subspaces in hermitean spaces of countable dimension. II

1985 ◽  
Vol 188 (3) ◽  
pp. 287-311
Author(s):  
Werner B�ni
Keyword(s):  
Countable Dimension

scholarly journals Topological characterizations of amenability and congeniality of bases

2020 ◽  
Vol 21 (1) ◽  
pp. 1
Author(s):  
Sergio R. López-Permouth ◽  
Benjamin Stanley
Keyword(s):  
Direct Product ◽  
Topological Spaces ◽  
Countable Dimension ◽  
One To One ◽  
Natural Way

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F<sup>B</sup>, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

scholarly journals On the stability of barrelled topologies, III

1980 ◽  
Vol 22 (1) ◽  
pp. 99-112 ◽  
Author(s):  
W.J. Robertson ◽  
I. Tweddle ◽  
F.E. Yeomans
Keyword(s):  
Sufficient Conditions ◽  
Vector Subspace ◽  
Finite Dimensional ◽  
Countable Dimension ◽  
The Stability ◽  
Bounded Sets ◽  
Barrelled Space

Let E be a barrelled space with dual F ≠ E*. It is shown that F has uncountable codimension in E*. If M is a vector subspace of E* of countable dimension with M ∩ F = {o}, the topology τ(E, F+M) is called a countable enlargement of τ(E, F). The results of the two previous papers are extended: it is proved that a non-barrelled countable enlargement always exists, and sufficient conditions for the existence of a barrelled countable enlargement are established, to include cases where the bounded sets may all be finite dimensional. An example of this case is given, derived from Amemiya and Kōmura; some specific and general classes of spaces containing a dense barrelled vector subspace of codimension greater than or equal to c are discussed.

scholarly journals Countable vector lattices

1974 ◽  
Vol 10 (3) ◽  
pp. 371-376 ◽  
Author(s):  
Paul F. Conrad
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Real Numbers ◽  
Vector Lattices ◽  
Direct Sum ◽  
Ordered Vector Spaces ◽  
Countable Dimension

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

scholarly journals On the stability of barrelled topologies, I

1979 ◽  
Vol 20 (3) ◽  
pp. 385-395 ◽  
Author(s):  
W.J. Robertson ◽  
F.E. Yeomans
Keyword(s):  
Dual Space ◽  
Vector Spaces ◽  
Mackey Topology ◽  
Topological Vector Spaces ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Countable Dimension ◽  
The Stability ◽  
Locally Convex ◽  
Barrelled Space

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.

scholarly journals Independence and consistency proofs in quadratic form theory

1991 ◽  
Vol 56 (4) ◽  
pp. 1195-1211 ◽  
Author(s):  
James E. Baumgartner ◽  
Otmar Spinas
Keyword(s):  
Quadratic Form ◽  
Set Theory ◽  
Orthogonal Group ◽  
Continuum Hypothesis ◽  
Infinite Dimension ◽  
Form Theory ◽  
Countable Dimension ◽  
Straight Lines ◽  
Consistency Proofs ◽  
The Continuum

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

Fr�chet differentiable norms on spaces of countable dimension

1992 ◽  
Vol 58 (5) ◽  
pp. 471-476 ◽  
Author(s):  
Jon Vanderwerff
Keyword(s):  
Countable Dimension

Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial

2006 ◽  
Vol 49 (2) ◽  
pp. 265-269 ◽  
Author(s):  
W. K. Nicholson ◽  
Y. Zhou
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Arbitrary Ring ◽  
Semisimple Module ◽  
Countable Dimension

AbstractIf C = C(R) denotes the center of a ring R and g(x) is a polynomial in C[x], Camillo and Simón called a ring g(x)-clean if every element is the sum of a unit and a root of g(x). If V is a vector space of countable dimension over a division ring D, they showed that end DV is g(x)-clean provided that g(x) has two roots in C(D). If g(x) = x – x2 this shows that end DV is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that end RM is g(x)-clean for any semisimple module M over an arbitrary ring R provided that g(x) ∈ (x – a)(x – b)C[x] where a, b ∈ C and both b and b – a are units in R.

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