Search results
Results From The WOW.Com Content Network
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 ...
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:
The following are equivalent: is (locally) bounded; [3] (Definition): maps bounded subsets of its domain to bounded subsets of its codomain; [3] maps bounded subsets of its domain to bounded subsets of its image := (); [3]
A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. 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.
Bounded set (topological vector space), a set in which every neighborhood of the zero vector can be inflated to include the set; Bounded variation, a real-valued function whose total variation is bounded; Bounded pointer, a pointer that is augmented with additional information that enable the storage bounds within which it may point to be deduced
is called uniformly bounded if there exists an element from and a real number such that ((),). Examples. Every uniformly convergent sequence of ...
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, a local ...