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Books > Computing & IT > General theory of computing
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Bornologies and Lipschitz Analysis (Hardcover)
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Bornologies and Lipschitz Analysis (Hardcover)
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This monograph, for the first time in book form, considers the
large structure of metric spaces as captured by bornologies:
families of subsets that contain the singletons, that are stable
under finite unions, and that are stable under taking subsets of
its members. The largest bornology is the power set of the space
and the smallest is the bornology of its finite subsets. Between
these lie (among others) the metrically bounded subsets, the
relatively compact subsets, the totally bounded subsets, and the
Bourbaki bounded subsets. Classes of functions are intimately
connected to various bornologies; e.g., (1) a function is locally
Lipschitz if and only if its restriction to each relatively compact
subset is Lipschitz; (2) a subset is Bourbaki bounded if and only
if each uniformly continuous function on the space is bounded when
restricted to the subset. A great deal of attention is given to the
variational notions of strong uniform continuity and strong uniform
convergence with respect to the members of a bornology, leading to
the bornology of UC-subsets and UC-spaces. Spaces on which its
uniformly continuous real-valued functions are stable under
pointwise product are characterized in terms of the coincidence of
the Bourbaki bounded subsets with a usually larger bornology.
Special attention is given to Lipschitz and locally Lipschitz
functions. For example, uniformly dense subclasses of locally
Lipschitz functions within the real-valued continuous functions,
Cauchy continuous functions, and uniformly continuous functions are
presented. It is shown very generally that a function between
metric spaces has a particular metric property if and only if
whenever it is followed in a composition by a real-valued Lipschitz
function, the composition has the property. Bornological
convergence of nets of closed subsets, having Attouch-Wets
convergence as a prototype, is considered in detail. Topologies of
uniform convergence for continuous linear operators between normed
spaces is explained in terms of the bornological convergence of
their graphs. Finally, the idea of a bornological extension of a
topological space is presented, and all regular extensions can be
so realized.
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