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Register a new userNormed Omega-Group
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Authors
Aleks Kleyn
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Since sum which is not necessarily commutative is defined in Omega-algebra A, then Omega-algebra A is called Omega-group. I also considered representation of Omega-group. Norm defined in Omega-group allows us to consider continuity of operations and continuity of representation.
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The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $Omega$-group. Theory of integration of maps into $Omega$-group has a lot of common with theory of integration of functions of real variable. However I had to change some statements, since they implicitly assume either compactness of range or total order in $Omega$-group.
We prove the Zabreikos lemma in 2-Banach spaces. As an application we shall prove a version of the closed graph theorem and open mapping theorem.
We modify the very well known theory of normed spaces $(E, orm)$ within functional analysis by considering a sequence $( orm_n : ninN)$ of norms, where $ orm_n$ is defined on the product space $E^n$ for each $ninN$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces; it is designed to relate to general spaces $L^p$ for $pin [1,infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a `multi-normed space $((E^n, orm_n) : nin N)$, where $(E, orm)$ is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum and maximum multi-norm based on a given space. Multi-norms measure `geometrical features of normed spaces, in particular by considering their `rate of growth. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces through `multi-null sequences, and to `multi-bounded linear operators, which are exactly the `multi-continuous operators. We define a new Banach space ${mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of `orthogonal decompositions of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual space.
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams which are Kauffmans reduction of the braid group algebra to the Temperley-Lieb algebra. We introduce an algorithm for computing all planer diagrams in a given dimension. The algorithm can also be used to multiply planer diagrams and find their matrix representation.
The aim of this note is to study existence and main properties of direct and inverse limits in the category of normed $L^0$-modules (in the sense of Gigli) over a metric measure space.
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