scholarly journals On the Notion of Scalar Product for Finite-dimensional Diffeological Vector Spaces

2018 ◽  
Vol 34 ◽  
pp. 18-27 ◽  
Author(s):  
Ekaterina Pervova
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Scalar Product ◽  
Vector Spaces ◽  
Initial Space ◽  
Finite Dimensional ◽  
Standard Space

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note, a way to dispense withthis issue is considered, by introducing a notion of pseudo-metric, which, said informally, is the least-degeneratesymmetric bilinear form on a given space. This notion is applied to make some observations on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in theobvious way, a smooth scalar product on the dual.

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 Some Invariants of a Bilinear Form

1961 ◽  
Vol 4 (3) ◽  
pp. 261-264
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Special Cases ◽  
Orthogonal Subspace

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

Lattice vector spaces and linear transformations

2019 ◽  
Vol 12 (02) ◽  
pp. 1950031
Author(s):  
Geena Joy ◽  
K. V. Thomas
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Invertible Matrix ◽  
Linear Transformations ◽  
Lattice Vector ◽  
Dimensional Lattice ◽  
Finite Dimensional

This paper introduces the concept of lattice vector space and establishes many important results. Also, this paper deals with linear transformations on lattice vector spaces and discusses their elementary properties. We prove that every finite dimensional lattice vector space is isomorphic to [Formula: see text] and show that the set of all columns (or the set of all rows) of an invertible matrix over [Formula: see text] is a basis for [Formula: see text].

scholarly journals Relative invariants and b-functions of prehomogeneous vector spaces

1985 ◽  
Vol 98 ◽  
pp. 139-156 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

Let G be a connected linear algebraic group, p a rational representation of G on a finite-dimensional vector space V, all defined over C.

Clifford and Heisenberg algebras

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Bilinear Form ◽  
Scalar Product ◽  
Heisenberg Algebra ◽  
Symmetric Bilinear Form ◽  
Rham Complex ◽  
Heisenberg Algebras ◽  
De Rham ◽  
Symplectic Vector Space

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then specializes the construction of the Clifford algebra to the case of V ⊕ V*. Next, the chapter argues that, if (V,ω‎) is a symplectic vector space, then the associated Heisenberg algebra is constructed and then specialized to the case of V ⊕ V*. Hereafter, the chapter considers the combination of the Clifford and Heisenberg algebras for V ⊕ V*, and constructs the complex Λ‎· (V*) ⊗ S· (V*), ̄ƌ) which is the subcomplex of polynomial forms in the de Rham complex. Finally, when V is equipped with a scalar product, this complex is related to a Witten complex over V.

Totally Variant Sets in Finite Groups and Vector Spaces

1968 ◽  
Vol 20 ◽  
pp. 701-710 ◽  
Author(s):  
Frederick Hoffman ◽  
Lloyd R. Welch
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Finite Groups ◽  
Dimensional Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Quaternion Group ◽  
Finite Dimensional ◽  
Identity Automorphism ◽  
Space Case

We are concerned here with the question of which finite groups and vector spaces possess subsets which are moved by every non-identity automorphism (in the vector-space case—non-singular linear transformation). We find that this is the case for all but four finite-dimensional vector spaces (2-, 3-, and 4-dimensional space over Z2, 2-dimensional space over Z3), and for all finite groups except for those corresponding to the vector-space exceptions, and the quaternion group of order eight. The question was first posed to the authors, in the vector-space case, by Morris Marx.

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 Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863.

2021 ◽  
Vol 29 (3) ◽  
pp. 117-127
Author(s):  
Kazuhisa Nakasho ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Algebraic Structure ◽  
Linear Independence ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Real Vector ◽  
Type Conversion ◽  
Finite Dimensional ◽  
Finite Sequences ◽  
Complicated Process

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

scholarly journals On the Index of a Symmetric Form

1960 ◽  
Vol 3 (3) ◽  
pp. 293-295
Author(s):  
Jonathan Wild
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Dimensional Vector ◽  
Symmetric Form ◽  
Characteristic P ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.

scholarly journals A classification of irreducible prehomogeneous vector spaces and their relative invariants

1977 ◽  
Vol 65 ◽  
pp. 1-155 ◽  
Author(s):  
M. Sato ◽  
T. Kimura
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Rational Representation ◽  
Finite Dimensional Vector Space ◽  
Prehomogeneous Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

LetGbe a connected linear algebraic group, andpa rational representation ofGon a finite-dimensional vector spaceV, all defined over the complex number fieldC.We call such a triplet (G, p, V) aprehomogeneous vector spaceifVhas a Zariski-denseG-orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces whenpis irreducible, and to investigate their relative invariants and the regularity.

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