![]() Let X X and Y Y be pointwise uniform convergence spaces. (People say âweak uniform convergenceâ in various contexts, but nobody ever says âpointwise uniform convergenceâ.) Subsidiary concepts Finally, âpointwiseâ was âweakâ in earlier versions of this page, until I changed it to have a more transparent meaning and avoid potential conflicts. On the other hand, âasymptoticâ conflicts with the meaning of asymptotic function ? in complexity theory (which is weaker, and in fact says that functions are asymptotic when their logarithms are asymptotic in our sense), although it matches the meaning of asymptote ? in elementary analytic geometry. ![]() That said, my use of âquasiââ follows quasimetric and quasiuniform space, so it at least is probably correct. ![]() Possibly other terms appear in the literature (besides the fact that âpointwiseâ may simply be left out). Terminology warningĪmong the terms defined (in boldface) above, I ( Toby Bartels) invented âasymptotic filterâ and those involving âpointwiseâ and âquasiââ, because no appropriate terms appeared in the reference that I used. The conditions (1,2,3,4,5,6,7) correspond respectively to the conditions (6,4,5,2,3,1,0) at uniform space (as of, in case the latter are ever renumbered). So really, it is simpler to include the improper filter among the asymptotic filters. It is traditional to consider only proper asymptotic filters (or equivalently to consider only asymptotic nets), which allows one to leave out (2) but in that case (5) and (6) must be modified to apply only when the generated filters are proper: (5) applies only when each element of F F meets each element of G G, and (6), if used, applies only when X X is inhabited. Assuming excluded middle, we may take G G to be F F itself (and take S S to be R R), rendering this condition trivial in classical mathematics.Ī (pointwise) (quasi)-uniform convergence structure/space that satisfies (7) may be called (quasi)- uniformly regular space|uniformly regular (although â(quasi)-uniformly locally decomposableâ would be more proper, since there is no reason why such a space should be regular, even in the symmetric case). ![]() This process is experimental and the keywords may be updated as the learning algorithm improves.Lim n â â â a n â b n â = 0. These keywords were added by machine and not by the authors. At the end of Section 9.2, we state and prove an important result due to Weierstrass, which in a simple form states that “any continuous function on can be uniformly approximated by polynomials.” Keywords Finally, we also discuss the Abel summability of series. At the end of the section, we also include some foundations for the study of summability of series, which is an attempt to attach a value to a series that may not converge, thereby generalizing the concept of the sum of a convergent series. ![]() In Section 9.2, we discuss a characterization for interchanging limit and integration signs, and interchange of limit and differentiation signs for uniform convergence of sequences and series of functions. In addition, we present characterizations for interchanging limit and integration signs in sequences of functions. Our particular emphasis in Section 9.1 is to present the definitions and simple examples of pointwise and uniform convergence of sequences. In this chapter we consider sequences and series of real-valued functions and develop uniform convergence tests, which provide ways of determining quickly whether certain sequences and infinite series have limit functions. ![]()
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