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Title:The Injective Envelope as the Space of Extremal Functions
Author(s):Dong, Zhou
Doctoral Committee Chair(s):Henson, C. Ward
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:2008
Description:88 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008.
Other Identifier(s):(MiAaPQ)AAI3347297
Date Available in IDEALS:2015-09-28
Date Deposited:2008

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