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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.
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 ...
The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds .) 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 .
A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
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 ...
A densely defined symmetric [clarification needed] operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That is, Tx|x ≥ −a ||x|| 2 for all x in the domain of T (or alternatively Tx|x ≥ a ||x|| 2 since a is arbitrary). [8] If both T and −T are bounded from below then T is ...
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant.
A locally bounded TVS is a TVS that possesses a bounded neighborhood of the origin. By Kolmogorov's normability criterion , this is true of a locally convex space if and only if the topology of the TVS is induced by some seminorm .