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A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
Andrew Yao showed [3] that there exists an efficient solution for range queries that involve semigroup operators. He proved that for any constant c, a pre-processing of time and space () allows to answer range queries on lists where f is a semigroup operator in (()) time, where is a certain functional inverse of the Ackermann function.
All the operators (except typeof) listed exist in C++; the column "Included in C", states whether an operator is also present in C. Note that C does not support operator overloading. When not overloaded, for the operators && , || , and , (the comma operator ), there is a sequence point after the evaluation of the first operand.
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 ...
An instance of a variable symbol is bound, in contrast, if the value of that variable symbol has been bound to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as ...
Any compact operator is strictly singular, but not vice versa. [6] A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (Schauder's theorem). [7] If : is bounded and compact, then: [5] [7] the closure of the range of is separable.
The closed range theorem, which says an operator (under some assumption) has closed image if and only if its transpose has closed image (see closed range theorem#Sketch of proof). The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section. What we have is: [9]
In computer programming, bounds checking is any method of detecting whether a variable is within some bounds before it is used. It is usually used to ensure that a number fits into a given type (range checking), or that a variable being used as an array index is within the bounds of the array (index checking).