analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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The term ‘bounded’ has several meaning in different branches of mathematics. For a general axiomatic approach to boundedness, see bornological set. Here we list definitions in various fields.
Let be a metric space. A subset is bounded if there is some real number such that for all .
This generalises immediately to pseudometric spaces, quasimetric spaces, extended metric spaces, and most generally to Lawvere metric spaces.
We can also generalise to gauge spaces:
Let be a gauge space. A subset is bounded if there is some real number such that for all and all gauging distances .
This generalises immediately to quasigauge spaces.
The family of all bounded sets of a quasigauge space (and hence of the more particular kinds of spaces above) defines a bornology on its underlying set.
Let be a LCTVS. A subset is bounded if whenever is a neighbourhood of then there is some real number such that .
The family of all bounded sets of a LCTVS defines a bornology on its underlying set.
Last revised on July 5, 2017 at 05:45:40. See the history of this page for a list of all contributions to it.