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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 ...
(Bounded)Continuous, where the respondent is presented with a continuous scale; A respondent's answer to an open-ended question is coded into a response scale afterward. An example of an open-ended question is a question where the testee has to complete a sentence (sentence completion item). [9]
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
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 length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ). In other words, the left bound of the interval I n {\displaystyle I_{n}} can only increase ( a n + 1 ≥ a n {\displaystyle a_{n+1}\geq a_{n}} ), and the right bound can only decrease ( b n + 1 ≤ b n ...
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 ...
A bounded operator T on a Banach space is invertible, i.e. has a bounded inverse, if and only if T is bounded below, i.e. ‖ ‖ ‖ ‖, for some >, and has dense range. Accordingly, the spectrum of T can be divided into the following parts: