#### Date of Award

2012

#### Document Type

Dissertation

#### Degree Name

M.S. in Mathematics

#### Department

Mathematics

#### First Advisor

Gerard Buskes

#### Second Advisor

Erwin Mina Diaz

#### Third Advisor

Gerard Buskes

#### Relational Format

dissertation/thesis

#### Abstract

A vector seminormed space is a triple consisting of a vector space, a Dedekind complete Riesz space, and a vector valued seminorm, called a vector seminorm, defined on the vector space and taking values in the Riesz space. The collection of vector seminormed spaces with suitably defined morphisms is shown to be a category containing finite products. A theory of vector seminorms on the tensor products of vector seminormed spaces is developed in analogy with the theory of tensor products of Banach spaces. Accordingly, a reasonable cross vector seminorm, or simply tensor seminorm, is defined such that a vector seminorm is a tensor seminorm if and only if it is in between the injective and projective vector seminorms, in analogy with the theory for normed spaces. Moreover, a theory of the complexification of a vector seminormed space is developed in analogy with the theory of complexification of a Banach lattice, and a class of vector seminorms, called admissible vector seminorms, is defined and shown to be equivalent to the class of reasonable cross vector seminorms on the complexification. Finally, the theory is applied to Dedekind complete Riesz spaces to demonstrate uniqueness of the complexification modulus, in the process yielding multiple formulae for the modulus.

#### Recommended Citation

Dever, John William, "Tensor Products Of Vector Seminormed Spaces" (2012). *Electronic Theses and Dissertations*. 669.

https://egrove.olemiss.edu/etd/669