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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 ...
[2] [3] Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. 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.
If an entire function is of bounded type in both the upper and the lower half-plane then it is of exponential type equal to the higher of the two respective "mean types" [2] (and the higher one will be non-negative). An entire function of order greater than 1 (which means that in some direction it grows faster than a function of exponential ...
A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R 2 constrained by two parabolic curves x 2 + 1 and x 2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).
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 .
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 ...