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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 .
Every uniformly convergent sequence of bounded functions is uniformly bounded. The family of functions () = defined for real with traveling through the integers, is uniformly bounded by 1. The family of derivatives of the above family, ′ = , is not uniformly bounded.
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
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.
Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation.
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
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:
Consider a sequence of real-valued continuous functions { f n } n ∈ N defined on a closed and bounded interval [a, b] of the real line. If this sequence is uniformly bounded and uniformly equicontinuous, then there exists a subsequence { f n k } k ∈ N that converges uniformly.