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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 Banach spaces, pointwise absolute convergence implies pointwise convergence, and normal convergence implies uniform convergence. For functions defined on a topological space, one can define (as above) local uniform convergence and compact (uniform) convergence in terms of the partial sums of the series.
The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.
In general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. [5] If f is of bounded variation, then its Fourier series converges everywhere. If f is additionally continuous, the convergence is uniform. [6]
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).
Guide to this index. To avoid excessive verbiage, note that each of the following types of objects is a special case of types preceding it: sets, topological spaces, uniform spaces, topological abelian groups (TAG), normed vector spaces, Euclidean spaces, and the real/complex numbers.
A set A of functions continuous between two topological spaces X and Y is topologically equicontinuous at the points x ∈ X and y ∈ Y if for any open set O about y, there are neighborhoods U of x and V of y such that for every f ∈ A, if the intersection of f[U] and V is nonempty, f[U] ⊆ O.