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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
Bounded poset, a partially ordered set that has both a greatest and a least element; Bounded set, a set that is finite in some sense Bounded function, a function or sequence whose possible values form a bounded set; Bounded set (topological vector space), a set in which every neighborhood of the zero vector can be inflated to include the set
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 .
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed.A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
be a family of functions indexed by , where is an arbitrary set and is the set of real or complex numbers. We call F {\displaystyle {\mathcal {F}}} uniformly bounded if there exists a real number M {\displaystyle M} such that
The set of all bounded sequences forms the sequence space. [ citation needed ] The definition of boundedness can be generalized to functions f : X → Y {\displaystyle f:X\rightarrow Y} taking values in a more general space Y {\displaystyle Y} by requiring that the image f ( X ) {\displaystyle f(X)} is a bounded set in Y {\displaystyle Y} .
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.}
If {x n} is bounded, then compactness of C implies that there exists a subsequence x nk such that C x nk is norm convergent. So x nk = (I - C)x nk + C x nk is norm convergent, to some x. This gives (I − C)x nk → (I − C)x = y. The same argument goes through if the distances d(x n, Ker(I − C)) is bounded. But d(x n, Ker(I − C)) must be ...