University of Illinois Urbana-Champaign

The Injective Envelope as the Space of Extremal Functions

Dong, Zhou

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https://hdl.handle.net/2142/86918

Description

Title
The Injective Envelope as the Space of Extremal Functions
Author(s)
Dong, Zhou
Issue Date
2008
Doctoral Committee Chair(s)
Henson, C. Ward
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In this thesis, we study the injective envelope of metric spaces by viewing it as the space of extremal functions as defined by Isbell. Extremal functions are also Kate˘tov functions, which satisfy two inequalities derived from the triangle inequality. One of these inequalities, along with a minimality requirement, is used to define the extremal functions. We compare the extremal functions to other classes of functions defined similarly using one of the two inequalities from the definition of Kate˘tov functions. We also consider separability of the space of extremal functions. We give a general method for generating uncountably many extremal functions from one extremal function satisfying certain inequalities on a sequence of ordered pairs. Then we prove non-separability of the space of extremal functions over some metric subspaces of finite dimensional real Banach spaces and some bounded metric spaces by constructing such an extremal function. Lastly, we discuss some connections with Melleray's work on separability of the space of Kate˘tov functions.
Graduation Semester
2008
Type of Resource
text
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http://hdl.handle.net/2142/86918

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