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

Metric topologies can be viewed as one-dimensional measures. This dissertation is a topological study of two-dimensional measures. Attention is focused on locally convex vector topologies on infinite dimensional real spaces. A nabla (referred to in the literature as a 2-norm) is the analogue of a norm which assigns areas to the parallelograms. Nablas are defined for the classical normed spaces and techniques are developed for defining nablas on arbitrary spaces. The work here brings out a strong connection with tensor and wedge products. Aside from the normable theory, it is shown that nabla topologies need not be metrizable or Mackey. A class of concretely given non-Mackey nablas on the ℓp and Lp spaces is introduced and extensively analyzed. Among other results it is found that the topological dual of ℓ₁ with respect to these nabla topologies is C₀, one of the spaces infamous for having no normed predual. Also, a connection is made with the theory of two-norm convergence (not to be confused with 2-norms). In addition to the hard analysis on the classical spaces, a duality framework from which to study the softer aspects is introduced. This theory is developed in analogy with polar duality. The ideas corresponding to barrelledness, quasi-barrelledness, equicontinuity and so on are developed. This dissertation concludes with a discussion of angles in arbitrary normed spaces and a list of open questions.

by Kin-Ming Liu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Bibliography: leaves 66-67. / Chapter §0 --- Introduction / Chapter §1 --- Preliminaries and notations / Chapter §2 --- A summary on ultra-(DF)-spaces and order-ultra-（DF)-spaces / Chapter §3 --- " ""Dual"" properties between projective and inductive topologies in topological vector spaces" / Chapter §4 --- Application of barrelledness on continuity of bilinear mappings and projective tensor product / Chapter §5 --- Countably order-quasiultrabarrelled spaces

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

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (leaves 295-307).

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 42-44). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1998. / Typescript. Vita. Includes bibliographical references (leaves 42-44). Also available on the Internet.

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by Yun-ming Tang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 92-93). / Chapter Chapter 1 --- Sequence Spaces / Chapter 1.1 --- Sequence spaces --- p.1 / Chapter 1.2 --- "Duality theorems on <λ, λX>" --- p.6 / Chapter 1.3 --- Topological properties of sequence spaces --- p.30 / Chapter 1.4 --- Diagonal maps --- p.41 / Chapter Chapter 2 --- Vector sequence spaces / Chapter 2.1 --- λ-summability of vector sequences --- p.48 / Chapter 2.2 --- "A duality theorem on <Λ(E),Λ(E)X>" --- p.62 / Chapter 2.3 --- "The topological duals of [λ[E],II(ρ,ξ)},(λ(E)(ρ,ξ)) and [ λw(E), B(ρ,ξ) ]" --- p.75 / Chapter 2.4 --- Fundamentally λ-bounded spaces --- p.86 / Reference --- p.92

本論文介紹Teichmuller 空間的參數化方法。我們會就此題目會作一歷史回顧，然後介紹Fenchel-Nielsen的參數化方法，最後集中討論在緊密且有界之曲面上的Teichmuller 空間以六角形分割之參數化方法。 / This thesis is an exposition of different parametrizations of the Teichmuller space. We will give a historical review on this subject, and in particular introduce the Fenchel-Nielsen coordinate. Our main focus would be the cellular decomposition method to parametrize the Teichmuller spaces of compact surface with boundary. / Detailed summary in vernacular field only. / Wong, Yun Shun Matthias. / "October 2012." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 42-43). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Teichmüller Space --- p.4 / Chapter 2.1 --- Definition of Teichmüller Space --- p.4 / Chapter 2.2 --- Historical review --- p.6 / Chapter 3 --- Fenchel-Nielsen Coordinate --- p.10 / Chapter 4 --- Cellular Decomposition Method --- p.18 / Chapter 4.1 --- Ideal Triangulation --- p.19 / Chapter 4.2 --- Edge Invariant --- p.22 / Chapter 4.2.1 --- E-Invariant of Colored Hexagon --- p.23 / Chapter 4.3 --- E-coordinate and its Generalization --- p.24 / Chapter 4.4 --- Edge Path and Edge Cycle --- p.26 / Chapter 4.5 --- Parametrization Theorems --- p.26 / Chapter 4.6 --- Discussion on the Results --- p.30 / Chapter 4.7 --- The Variational Proof of Theorem 4.8 --- p.32 / Chapter 4.7.1 --- Overview of the Proof --- p.32 / Chapter 4.7.2 --- The Energy Function on Hexagon --- p.35 / Chapter 4.7.3 --- The Energy Function on Length Structure and the Proof of Theorem 4.8 --- p.37 / Bibliography --- p.42