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A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P. A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest ...
A family of subsets of a topological vector space is said to be uniformly bounded in , if there exists some bounded subset of such that , which happens if and only if is a bounded subset of ; if is a normed space then this happens if and only if there exists some real such that ‖ ‖.
A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, E replaces the "size" ε, and a subset is of size E if its Cartesian square is a subset of E.) [4]
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 .
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.).
Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. [1] The supremum (abbreviated sup; pl.: suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of , if such an element exists. [1]
A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of ...
The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a topological space , and Σ is the sigma-algebra of Borel sets in X , then r c a ( X ) {\displaystyle rca(X)} is the subspace of c a ( Σ ) {\displaystyle ca(\Sigma )} consisting of all regular Borel measures on X .