scholarly journals A characterization of determinants over topological fields

1974 ◽  
Vol 15 (1) ◽  
pp. 60-62 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Vector Space ◽  
Surjective Homomorphism ◽  
Space Topology ◽  
Topological Field ◽  
Topological Fields ◽  
Vector Space Topology

Let K be a topological field. On introducing the vector-space topology on the n × n matrices over K, it becomes clear that the determinant map φ enjoys the following properties:(A) φ is a continuous surjective homomorphism from GLn(K) to K*,(B) φ(μa) = μn φ(a) for each non-zero φ in K, and all a in GLn(K).

scholarly journals Periodic Boehmians

1989 ◽  
Vol 12 (4) ◽  
pp. 685-692 ◽  
Author(s):  
Dennis Nemzer
Keyword(s):  
Vector Space ◽  
Unit Circle ◽  
Topological Vector Space ◽  
Fourier Coefficients ◽  
Growth Conditions ◽  
Generalized Functions ◽  
Space Topology ◽  
Vector Space Topology

A class of generalized functions, called periodic Boehmians, on the unit circle, is studied. It is shown that the class of Boehmians contain all Beurling distributions. An example of a hyperfunctlon that is not a Boehmian is given. Some growth conditions on the Fourier coefficients of a Boehmian are given. It is shown that the Boehmians, with a given complete metric topological vector space topology, is not locally bounded.

scholarly journals On a characterization of path connected topological fields

2019 ◽  
Vol 223 (12) ◽  
pp. 5279-5284
Author(s):  
Xavier Caicedo ◽  
Guillermo Mantilla-Soler
Keyword(s):  
Path Connected ◽  
Topological Fields

scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

scholarly journals The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View

2005 ◽  
Vol 12 (03) ◽  
pp. 207-229 ◽  
Author(s):  
Gen Kimura ◽  
Andrzej Kossakowski
Keyword(s):  
Vector Space ◽  
Quantum State ◽  
Point Of View ◽  
Vector Spaces ◽  
Bloch Vector ◽  
State Spaces ◽  
Spherical Coordinate ◽  
N Level ◽  
Dual Property

Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU (N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Moduli Space ◽  
Deformation Theory ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Exterior Degree ◽  
Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

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