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Thesis (Ph. D.)--University of Wisconsin, 1964. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record.

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The basis for this thesis is H. S. Wall's book, Creative Mathematics, with particular emphasis on the chapter in that book entitled "More About Linear Spaces."

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf. 29.

Diss. Uppsala : Univ., 2002.

In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.

We begin by constructing a vector space over a field F , which we call a (G,σ) space of the set W = V₁xV₂... xVn , a cartesian product, where Vi is a finite-dimensional vector space over an arbitrary field F , G is a subgroup of the full symmetric group Sn and σ is a linear character of G . This space generalizes the spaces called the symmetry class of tensors defined by Marcus and Newman [1]. We can obtain the classical spaces, namely the Tensor space, the Grassman space and the symmetric space, by particularizing the group G and the linear character σ in our (G,σ) space. If (v₁,v₂,..., vn ) ∈ W , we shall denote the "decomposable" element in our space by v₁Δv₂…Δvn and call it the (G,σ) product or the Pure product if there is no confusion regarding G and σ, of the vectors v₁,v₂,..., vn . This corresponds to the tensor product, the skew symmetric product and the symmetric product in the classical spaces. The purpose of this thesis is to determine a necessary and sufficient condition for the vanishing of the (G,σ) product of the vectors v₁,v₂,..., vn in the general case. The results for the classical spaces are well-known and are deduced from our main theorem. We use the "universal mapping property" of the (G,σ) space to prove the necessity of our condition. These conditions are stated in terms of determinant-like functions of the matrices associated with the set of vectors v₁,v₂,...,vn. / Science, Faculty of / Mathematics, Department of / Graduate

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This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.