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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 .
A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
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.).
A real-valued or complex-valued function defined on some topological space is called a locally bounded functional if for any there exists a neighborhood of such that () is a bounded set. That is, for some number M > 0 {\displaystyle M>0} one has | f ( x ) | ≤ M for all x ∈ A . {\displaystyle |f(x)|\leq M\quad {\text{ for all }}x\in A.}
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval [0,1] such that for every point : there is a neighbourhood of x {\displaystyle x} where all but a finite number of the functions of R {\displaystyle R} are 0, and
Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem. A metric space is separable if and only if it is homeomorphic to a totally bounded metric space. [3] The closure of a totally bounded subset is again totally bounded. [6]
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.