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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
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 Helly. A more general version of the theorem asserts compactness of the space BV loc of functions locally of bounded total variation that are uniformly bounded at a point.
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 set of discontinuities of a regulated function of bounded variation BV is countable for such functions have only jump-type of discontinuities. To see this it is sufficient to note that given ϵ > 0 {\displaystyle \epsilon >0} , the set of points at which the right and left limits differ by more than ϵ {\displaystyle \epsilon } is finite.
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions. This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence () of time partitions: [1]
The essential difference between this and other well-known moment problems is that this is on a half-line [0, ∞), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−∞, ∞).
As a corollary, every uniformly bounded equicontinuous sequence in C(X) contains a subsequence that converges uniformly to a continuous function on X. In view of Arzelà–Ascoli theorem, a sequence in C ( X ) converges uniformly if and only if it is equicontinuous and converges pointwise.