# Continuity In Banach Spaces

Burns, J (2014) Continuity In Banach Spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

 Preview
PDF
Primary Text

## Abstract

The main theme of this document and much of the author's research so far is to use porosity (as well as category) to describe how typical a variety of continuity conditions are within certain collections of functions. The intuition behind the work is that ``most things should behave according to a pattern or principle of action''. The pattern that the author has in mind is that things should: flow, be localized, and oscillate. Indeed, this theme occurs throughout the authors included work. As a side note: the reader with a very particular background and side interest may recognize these three principles as Yu-Won-Hwa, an observation that will help with understanding future naming conventions. In Chapter 1, we study what properties a typical bounded real valued derivative possesses, in terms of continuity. We first prove results for finite dimensional domains. Additionally, we obtain some results when the domain is a subset of a general Banach space with a Frechet differentiable norm. In Chapter 2, we study porosity in relation to bounded variation. In particular, we show that when we suitably norm the space of functions with bounded variation, then the Cantor function becomes the typical example of a function in that space. In Chapter 3, we study how typical (in the sense of both category and porosity) it is for a function that is twice partial differentiable to have equal mixed partial derivatives. As it turns out, the ability to satisfy Clairaut's Theorem is infrequent. In Chapter 4, we study a general condition that implies we have an open, dense, co-porous set whenever we are looking at a set defined by a seminorm in a particular way. This allows us to prove a number of unifying results (without having to individually go through many cases involving much calculation). In Chapter 5, we introduce a few open questions that the author has recently been working on directly, or has formulated for further study. In the Appendix, we introduce the concept of porosity in an easy to follow format, with illustrative diagrams to guide the reader in their pursuit of intuition.

## Share

Citation/Export: Select format... Citation - Text Citation - HTML Endnote BibTex Dublin Core OpenURL MARC (ISO 2709) METS MODS EP3 XML Reference Manager Refer

## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Burns, Jjwb19@pitt.eduJWB19
ETD Committee:
Committee ChairLennard, Clennard@pitt.eduLENNARD
Committee MemberBeatrous, Fbeatrous@pitt.eduBEATROUS
Committee MemberCaginalp, Gcaginalp@pitt.eduCAGINALP
Committee MemberTurett, Bturett@oakland.edu
Date: 17 September 2014
Date Type: Publication
Defense Date: 2 August 2014
Approval Date: 17 September 2014
Submission Date: 7 July 2014
Access Restriction: 5 year -- Restrict access to University of Pittsburgh for a period of 5 years.
Number of Pages: 131
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Functional Analysis, Analysis, Continuity, Banach space, Derivatives
Date Deposited: 17 Sep 2014 17:28
URI: http://d-scholarship.pitt.edu/id/eprint/22665

### Available Versions of this Item

• Continuity In Banach Spaces. (deposited 17 Sep 2014 17:28) [Currently Displayed]