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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 ...
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 .
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 ...
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]
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.).
The seminorm p w (x) for w positive in the predual is defined to be B(w, x * x) 1/2. If B is a vector space of linear maps on the vector space A, then σ(A, B) is defined to be the weakest topology on A such that all elements of B are continuous. The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x ...
Recall that a linear map is bounded if and only if it maps any sequence converging to in the domain to a bounded subset of the codomain. [4] In particular, any linear map that is sequentially continuous at the origin is bounded. Every bounded linear operator from into a seminormed space is continuous. [4]