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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 ...
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 ...
In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant. This definition can also be extended to the case when the functions in the family U take values in some metric space, by again replacing the absolute value with the distance function.
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.
Maximum point-blank range is principally a function of a cartridge's external ballistics and target size: high-velocity rounds have long point-blank ranges, while slow rounds have much shorter point-blank ranges. Target size determines how far above and below the line of sight a projectile's trajectory may deviate.
An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite ...
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 ...
It may be assumed that all variables are bounded from below and above: such bounds either appear naturally in the application at hand, or can be enforced without losing any optimal solutions. But, even with linear objective functions the problem is NP-hard and hence presumably cannot be solved in polynomial time.