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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.
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 measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions.It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian mathematician and geometer, who published independent proofs respectively ...
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.
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]
Pointwise Cauchy-convergence; Uniform Cauchy-convergence; Implications are cases of earlier ones, except: - Uniform convergence both pointwise convergence and uniform Cauchy-convergence. - Uniform Cauchy-convergence and pointwise convergence of a subsequence uniform convergence.
The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals on a set of finite measure converge to the integral of the limit function. If the convergence were uniform this would be a trivial result, and Littlewood's third principle tells us that the convergence is almost uniform, that is ...