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A real-valued function is bounded if and only if it is bounded from above and below. [ 1 ] [ additional citation(s) needed ] An important special case is a bounded sequence , where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers .
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family.
Any such M is called an upper bound. Likewise, if, for some real m, a n ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded.
Corollary — If a sequence of bounded operators () converges pointwise, that is, the limit of (()) exists for all , then these pointwise limits define a bounded linear operator . The above corollary does not claim that T n {\displaystyle T_{n}} converges to T {\displaystyle T} in operator norm, that is, uniformly on bounded sets.
Note that this more general concept of boundedness does not correspond to a notion of "size". A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly.
In a sense, this means that the sequence must be bounded, with a bound that gets smaller as the sample size increases. This suggests that if a sequence is o p ( 1 ) {\displaystyle o_{p}(1)} , then it is O p ( 1 ) {\displaystyle O_{p}(1)} , i.e. convergence in probability implies stochastic boundedness.
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed.A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
As particular examples of Banach spaces, Dunford & Schwartz (1958, Chapter IV) consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x = ( x i ) of real or complex numbers is defined by