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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 ...
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.
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.
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
Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, one might define point B to be between two other points A and C, if the distance | AB | added to the distance | BC | is equal to the distance | AC |.
Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
Each set has a supremum (infimum), if it is bounded from above (below). Proof: Without loss of generality one can look at a set A ⊂ R {\displaystyle A\subset \mathbb {R} } that has an upper bound. One can now construct a sequence ( I n ) n ∈ N {\displaystyle (I_{n})_{n\in \mathbb {N} }} of nested intervals I n = [ a n , b n ] {\displaystyle ...