Let U, V be two vector spaces of dimensions n and m, respectively, over an algebraically closed field F; let U⊗V be their tensor product; and let Rk(U⊗V) be the set of all rank k tensors in U⊗V, that is Rk(U⊗V) = {[formula omitted] are each linearly independent in U and V respectively}. We first obtain conditions on two vectors X and Y that they be members of a subspace H contained in Rk(U⊗V). In chapter 2, we restrict our consideration to the rank 2 case, and derive a characterization of subspaces contained in R2(U⊗V). We show that any such subspace must be one of three types, and we find the maximum dimension of each type. We also find the dimension of the intersection of two subspaces of different types. Finally, we show that any maximal subspace has a dimension which depends only on its type. / Science, Faculty of / Mathematics, Department of / Graduate

Let U⊗V be a tensor product space over an algebraically closed field F ; let dim U = m and dim V = n ; let T be a linear transformation on U⊗V such that T preserves rank two tensors. We show that T preserves rank one tensors and this enables us to characterize T for all values of m and n. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we present a theory of cylinder measures from the viewpoint of inverse systems of measure spaces. Specifically, we consider the problem of finding limits for the inverse system of measure spaces determined by a cylinder measure μ over a vector space X. For any subspace Ω of the algebraic dual X* such that (X,Ω) is a dual pair, we establish conditions on μ which ensure the existence of a limit measure on Ω . For any regular topology G on Ω, finer than the topology of pointwise convergence, we give a necessary and sufficient condition on μ for it to have a limit measure on Ω Radon with respect to G We introduce the concept of a weighted system in a locally convex space. When X is a Hausdorff, locally convex space, and Ω is the topological dual of X , we use this concept in deriving further conditions under which μ will have a limit measure on Ω Radon with respect to G. We apply our theory to the study of cylinder measures over Hilbertian spaces and ℓ(ρ)-spaces, obtaining significant extensions and clarifications of many previously known results. / Science, Faculty of / Mathematics, Department of / Graduate

The object of this thesis is to examine properties of an abstract vector space of finite dimension n. The properties of the set of complex numbers are assumed, and the definition of a field and of an abelian group are not stated, although reference to these systems is made.

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A Dissertation Submitted of the Faculty of Science, University of the Witwatersrand, Johannesburg in Partial Fulfilment of the Requirements for the Degree of Master of Science / Let 4 be the class of pairs of loc ..My onvex spaces (X,V) “h ‘ch are such that every closed graph linear ,pp, 1 from X into V is continuous. It B is any class of locally . ivex l.ausdortf spaces. let & w . (X . (X.Y) e 4 for ,11 Y E B). " ‘his expository dissertation, * (B) is investigated, firstly i r arbitrary B . secondly when B is the class of C,-complete paces and thirdly whon B is a class of locally convex webbed s- .ces

Thesis (M. Sc. (Mathematics))--Rhodes University, 2002.

A non-trivial example of an L-topological space, the fuzzy real line is examined. Various L-topological properties and their relationships are developed. Extension theorems on the L-fuzzy real line as well as extension theorems on more general L-topological spaces follow. Finally, a theory of L-fuzzy vector spaces leads up to a fuzzy version of the Hahn-Banach theorem.

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