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A mathematical function the set of whose values is bounded. A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics, a function defined on some set with real or complex values ...
The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value. [6]
Definition of uniform continuity. is called uniformly continuous if for every real number there exists a real number such that for every with , we have . The set for each is a neighbourhood of and the set for each is a neighbourhood of by the definition of a neighbourhood in a metric space.
v. t. e. In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. [1]
Uniform boundedness. 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.
The test is as follows. Let {g n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g n+1 (x) ≤ g n (x) for all x ∈ E and positive integers n, and let {f n} be a sequence of real-valued functions such that the series Σf n (x) converges uniformly on E. Then Σf n (x)g n (x) converges uniformly on E.
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In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. [1] More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis ...