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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 ...
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.
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 .
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 union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
The description then gives distance, direction and various boundary descriptions as if one were walking the bounds pacing off the distance to the next corner where there is a change of direction. Where watercourses form part of the bounds their meander is generally taken as a straight line between the established corners and their monuments.
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.