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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 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 best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists. [5] [6] [7] Because of the integration by part formula, the integral exists also if the condition on f and g are inversed, that is, if f is of bounded variation and g is continuous.
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.
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.
More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if {f n} is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with ...
1.1 Bounded variation. 8 comments. 1.2 elliptical tube cuts a plan. 7 comments. 1.3 functions. 5 comments. 1.4 Equation solving. 6 comments. 1.5 test edit. 7 comments.