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 Cross-Ratio in Real Vector Space

2019 ◽  
Vol 27 (1) ◽  
pp. 47-60
Author(s):  
Roland Coghetto
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Cross Ratio ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Complex Vector ◽  
The Real ◽  
Mizar Mathematical Library ◽  
The Cross

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

Topologies in vector lattices

1952 ◽  
Vol 48 (4) ◽  
pp. 533-546 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Banach Lattices ◽  
Vector Spaces ◽  
Von Neumann ◽  
Topological Vector Spaces ◽  
Vector Lattices

This paper is concerned with topologies for vector lattices (in the sense of (1)) or spaces that satisfy the axioms K 1, 2, 3′ and 4 of (5) that are given by neighbourhoods. The notions of convergence introduced by Kantorovitch(5) and Birkhoff(1) do not in general lead to topologies which can be denned in terms of open sets, as, even for directed systems, the notion of closure derived from them is not such that the closure of every set is closed. In order to ensure this Kantorovitch introduces an axiom (his K6) which excludes even some Banach lattices. These notions of convergence are based more on the lattice aspect than the vector-space aspect of a vector lattice. In this paper the reverse is true, and it deals essentially with the application of the theory of topological vector spaces, as developed by von Neumann(9), Mackey (6, 7) and others, to vector lattices.

scholarly journals FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

2016 ◽  
Vol 101 (2) ◽  
pp. 277-287
Author(s):  
AARON TIKUISIS
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Space Structure ◽  
Vector Spaces ◽  
Ordered Vector Space ◽  
Finite Dimensional ◽  
Ordered Vector Spaces

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$.

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

scholarly journals Nonsmooth analysis and optimization on partially ordered vector spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Vector Lattice ◽  
Vector Spaces ◽  
Range Space ◽  
Topological Vector Spaces ◽  
Complete Vector Lattice ◽  
Lipschitz Mappings ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Complete Vector ◽  
Locally Convex

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

scholarly journals The deligne complex of a real arrangement of hyperplanes

1993 ◽  
Vol 131 ◽  
pp. 39-65 ◽  
Author(s):  
Luis Paris
Keyword(s):  
Vector Space ◽  
Finite Family ◽  
Real Vector Space ◽  
Real Vector ◽  
Arrangement Of Hyperplanes

Let V be a real vector space. An arrangement of hyperplanes in V is a finite family of hyperplanes of V through the origin. We say that is essential if ∩H ∊H = {0}

scholarly journals Axioms for convexity

1993 ◽  
Vol 47 (2) ◽  
pp. 179-197 ◽  
Author(s):  
W.A. Coppel
Keyword(s):  
Vector Space ◽  
Convex Sets ◽  
Real Vector Space ◽  
Real Vector ◽  
Complete Set

The basic elementary results about convex sets are derived successively from various properties of segments. The complete set of properties is shown to form a natural set of axioms characterising the convex sets in a real vector space.

scholarly journals New construction of algebras as quotients

2020 ◽  
Vol 5 (2) ◽  
pp. 53
Author(s):  
José Ramón Játem Lásser
Keyword(s):  
Vector Space ◽  
Natural Number ◽  
Clifford Algebras ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Natural Numbers ◽  
New Approach ◽  
Binary Operations ◽  
Direct Sum ◽  
New Construction

  In this article we have presented a new approach to define algebras using for a natural number k ≥ 2, the set of natural numbers in base k, none of their digits equal to zero. The study was developed in the context of vector R -spaces and the vector space definitions of the formal multiples of any element x of the field R, of the direct sum of vector spaces and binary operations on vector spaces were used. The results obtained were the construction of a vector space denoted by V, on the basis of the particular set of natural numbers in base k mentioned, which allowed novel ways of defining the well-known and very important algebras of complex numbers and that of quaternions on R as quotients of ideals of V, for suitably chosen ideals I. With this new approach and with the help of the vector spaces V, known algebras can be presented in a different way than those found up to now, by using certain ideals of those spaces in their quotient form. The spaces V can be over any field K and other algebras such as Clifford algebras can be constructed using this procedure.   Keywords: Algebras, Quotients in algebras, Complex numbers and quaternions as quotients of algebras.   Abstract En este artículo se ha presentado un nuevo enfoque para definir álgebras usando para un número natural k ≥ 2, el conjunto de números naturales en base k, ninguno de sus dígitos iguales a cero. El estudio se desarrolló en el contexto de los R-espacios vectoriales y se usaron las definiciones de espacio vectorial de los múltiplos formales de un elemento cualquiera x del cuerpo R, de la suma directa de espacios vectoriales y operaciones binarias sobre espacios vectoriales. Los resultados obtenidos fueron la construcción de un espacio vectorial denotado por V, sobre la base del particular conjunto de números naturales en base k mencionado, que permitió novedosas formas de definir las conocidas y muy importantes álgebras de los números complejos y la de los cuaterniones sobre R como cocientes de ideales de V, para ideales I convenientemente elegidos. Con este nuevo enfoque y con la ayuda de los espacios vectoriales V se pueden presentar álgebras conocidas de manera distinta a las encontradas hasta ahora, al usar en su forma de cociente ciertos ideales de esos espacios V. Los espacios V pueden ser sobre cualquier cuerpo K y otras álgebras como las álgebras de Clifford se pueden construir usando este procedimiento.   Palabras claves: Algebras, cocientes en álgebras, Números complejos y quaterniones como cocientes en álgebras.  

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