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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 ]
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
explicitly, for every Mackey convergent null sequence () = in (), the sequence (()) = is bounded. a sequence = = is said to be Mackey convergent to 0 if there exists a divergent sequence = = of positive real number such that the sequence () = is bounded; every sequence that is Mackey convergent to 0 necessarily converges to the origin (in the ...
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.
For example, any orthonormal sequence {f n} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {x n} is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). [78]
explicitly, for every Mackey convergent null sequence () = in (), the sequence (()) = is bounded; a sequence = = is said to be Mackey convergent to the origin if there exists a divergent sequence = = of positive real numbers such that the sequence () = is bounded; every sequence that is Mackey convergent to the origin necessarily converges to ...
In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance r/2 of s, and any two terms of the original sequence are within distance r/2 of ...
A region V bounded by the surface = with the surface normal n Suppose V is a subset of R n {\displaystyle \mathbb {R} ^{n}} (in the case of n = 3, V represents a volume in three-dimensional space ) which is compact and has a piecewise smooth boundary S (also indicated with ∂ V = S {\displaystyle \partial V=S} ).