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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.
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.
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard ...
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.
In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem , which states that a subset of C ( X ), the space of continuous functions on a compact Hausdorff space X , is compact if and only if it is closed, pointwise bounded and equicontinuous.
The theorem states that each infinite bounded sequence in has a convergent subsequence. [1] An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded . [ 2 ]
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence.