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The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
For example, an infinite set equipped with the discrete metric is bounded but not totally bounded: [3] every discrete ball of radius = / or less is a singleton, and no finite union of singletons can cover an infinite set.
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
2 Examples. 3 References. ... a uniformly bounded family of functions is a family of bounded ... where is an arbitrary set and is the set of real or complex numbers ...
The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. [39] If S {\displaystyle S} is a subset of a TVS X {\displaystyle X} such that every sequence in S {\displaystyle S} has a cluster point in S {\displaystyle S} then S {\displaystyle S} is totally bounded.
A locally bounded TVS is a TVS that possesses a bounded neighborhood of the origin. By Kolmogorov's normability criterion , this is true of a locally convex space if and only if the topology of the TVS is induced by some seminorm .
Bornology originates from functional analysis.There are two natural ways of studying the problems of functional analysis: one way is to study notions related to topologies (vector topologies, continuous operators, open/compact subsets, etc.) and the other is to study notions related to boundedness [2] (vector bornologies, bounded operators, bounded subsets, etc.).