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The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick ...
For a list of modes of convergence, see Modes of convergence (annotated index) Each of the following objects is a special case of the types preceding it: sets , topological spaces , uniform spaces , topological abelian group , normed spaces , Euclidean spaces , and the real/complex numbers.
Modes of convergence (annotated index) Mosco convergence; N. Normal convergence; P. Pointwise convergence; R. Radius of convergence; Convergence of random variables; S.
List of topologies – List of concrete topologies and topological spaces; Modes of convergence – Property of a sequence or series; Operator norm – Measure of the "size" of linear operators; Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.