## Electronic Theses and Dissertations

## Date

2019

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy

## Department

Mathematical Sciences

## Committee Chair

Maria Botelho

## Committee Member

Ben McCarty

## Committee Member

Irena Lasiecka

## Committee Member

Anna Kaminska

## Abstract

In this work, we consider different notions of openness for several operations on a large collection of different settings. We start with the scalar multiplication on sequence spaces, spaces of continuous functions, and integrable functions. We apply different techniques to derive weak-openness of multiplication on spaces of differentiable functions, endowed with a large collection of quasi-algebra norms. We consider the openness property of the dense-defined standard multiplication on spaces of integrable functions, as multiplication on these settings is only defined on a dense set. We adapt the definition of openness for the multiplication to include dense-defined products and then prove that the multiplication on these spaces, restricted to their corresponding domains, is uniformly open. We investigate sufficient and necessary conditions for pairs of functions to be points of local openness for the multiplication on normed spaces of differentiable functions. Next, we establish openness for other kinds of operations, more precisely, the maximum and minimum operations on spaces of scalar-valued integrable functions from a topological space into a finite measure on an algebra of subsets of that space. The last section of this work deals with the connection between openness of the multiplication on spaces of continuous functions and topological properties of the domain of those functions.

## Library Comment

Dissertation or thesis originally submitted to ProQuest

## Recommended Citation

Renaud, Holly Michelle, "Topological Properties of the Standard Operations on Spaces of Continuous Functions and Integrable Functions" (2019). *Electronic Theses and Dissertations*. 2735.

https://digitalcommons.memphis.edu/etd/2735

## Comments

Data is provided by the student.