scholarly journals A New Proof of the Pythagorean Theorem and Its Application to Element Decompositions in Topological Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Fred Greensite
Keyword(s):  
Vector Space ◽  
Algebraic Structure ◽  
Geometric Mean ◽  
Topological Algebra ◽  
Space Structure ◽  
Pythagorean Theorem ◽  
Euclidean Norm ◽  
Topological Algebras

We present a new proof of the Pythagorean theorem which suggests a particular decomposition of the elements of a topological algebra in terms of an “inverse norm” (addressing unital algebraic structure rather than simply vector space structure). One consequence is the unification of Euclidean norm, Minkowski norm, geometric mean, and determinant, as expressions of this entity in the context of different algebras.

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.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

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 $\mathbb{Q}$.

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

scholarly journals FOCK SPACE STRUCTURE FOR THE SIMPLEST PARASUPERSYMMETRIC SYSTEM

2001 ◽  
Vol 16 (15) ◽  
pp. 963-971 ◽  
Author(s):  
WEIMIN YANG ◽  
SICONG JING
Keyword(s):  
Vector Space ◽  
State Vector ◽  
Fock Space ◽  
Space Structure ◽  
The State ◽  
Degree Of Freedom ◽  
Basis Vectors

Structure of the state-vector space for a system consisting of one mode para-Bose and one mode para-Fermi degree of freedom with the same parastatistics order p is studied and a complete, orthonormal set of basis vectors in this space is constructed. There is an intrinsic double degeneracy for state vectors with m parabosons and n parafermions, where m ≠ 0, n ≠ 0 and n ≠ p. It is also shown that the degeneracy plays a key role in realization of exact supersymmetry for such a system.

scholarly journals On totally bounded universal algebras

2012 ◽  
Vol 21 (2) ◽  
pp. 151-165
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
INA D. CIOBANU ◽  
Keyword(s):  
Topological Algebra ◽  
Continuous Extension ◽  
Topological Algebras ◽  
Universal Algebras ◽  
Totally Bounded ◽  
General Properties

In the class of topological algebras of a given signature the notions of totally boundedness and a-pseudocompactness are introduced. A topological algebra is totally bounded if it is a subalgebra of a compact algebra. The general properties of totally bounded algebras are studied. The compactifications of topological algebras are investigated too. In particular, the problem of the continuous extension of the operation on the Stone-Cech compactification is studied.

scholarly journals On Uniform Topological Algebras

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
M. El Azhari
Keyword(s):  
Banach Algebra ◽  
Topological Algebra ◽  
Uniform Norm ◽  
Commutative Banach Algebra ◽  
Normed Algebra ◽  
Topological Algebras

Abstract The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M(A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

The learning of the vector space structure by sixth grade students

1971 ◽  
Vol 4 (2) ◽  
pp. 166-181 ◽  
Author(s):  
William E. Lamon ◽  
Leslie E. Huber
Keyword(s):  
Vector Space ◽  
Sixth Grade ◽  
Space Structure ◽  
Sixth Grade Students

Infrasequential Topological Algebras

1979 ◽  
Vol 22 (4) ◽  
pp. 413-418 ◽  
Author(s):  
T. Husain
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Convex ◽  
Locally Convex Algebra

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

Varieties Obeying Homotopy Laws

1977 ◽  
Vol 29 (3) ◽  
pp. 498-527 ◽  
Author(s):  
Walter Taylor
Keyword(s):  
Algebraic Structure ◽  
Topological Structure ◽  
Homotopy Group ◽  
Topological Algebra ◽  
Complete Analysis ◽  
Or Groups ◽  
Group Identities ◽  
The Many

The algebraic structure of a topological algebra influences its topological structure in a way which is profound but not well understood. (See § 7 below for various examples.) Here we examine this influence rather generally, and give a fairly complete analysis of one of the many forms it can take, namely, the influence of the identities of on the group identities obeyed by the homotopy group (or groups of the components) of .

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