The Injective Envelope as the Space of Extremal Functions

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Dong, Zhou

Contributors dc:contributor

• Henson, C. Ward

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3347297

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86918