This concept is often contrasted with uniformconvergence.

Uniformconvergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces.

Pointwise convergence may also be formulated as convergence in the topology which arises from the seminorm given by

Learn more about Real analysis in the online encyclopedia.(Site not responding. Last check: 2007-10-07)

The concept of convergence, central to analysis, is introduced via limits of sequences.

At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions.

The notion of uniformconvergence is important in this context.

Uniforms To You(Site not responding. Last check: 2007-10-07)

Modern uniforms are worn by armed forces and paramilitary organisations such as police, emergency services, security guards, in some workplaces and schools, and by inmates in prisons.

Uniformspaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniformcontinuity and uniformconvergence.

The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "''x'' is as close to ''a'' as ''y'' is to ''b''", while in a topological space you can only formalize "''x'' is as close to ''a'' as ''y'' is to ''a''".

In this module, metrics provide a language for discussing convergence and completeness, and so the emphasis is on the definition of a metric space and giving plenty of useful examples of metrics.

www.maths.ed.ac.uk /~carbery/FoA.html (932 words)

Uniform Convergence(Site not responding. Last check: 2007-10-07)

We investigate this convergence graphically by looking at the maximum difference between the partial sums and the function over the entire interval as a function of n.

Another way to visualize uniformconvergence is to look at a strip of width epsilon around the function.

Here we do not expect uniformconvergence of the partial sums to the function and in fact, we see that the partial sums, while they do converge pointwise, do not converge uniformly to the function.

We haven't yet had a formal definition of convergence for anything but sequences of real numbers, but the analogy with the real case will be pretty strong in these examples.

We'll come to a formal definition of convergence which is broad enough to encompass all of these examples, in the next section.

The next big step is to introduce a general mathematical setup for studying convergence and continuity which is broad enough to encompass the fairly diverse types of examples we have just been looking at, together with many more that will arise.

Uniform convergence of martingales in the branching random walk(Site not responding. Last check: 2007-10-07)

The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper.

This will imply that, on that set, the martingale limit \Wl is actually an analytic function of \la. The uniformconvergence results are used to obtain extensions of known results on the growth of $\Zn(nc+D)$ with $n$, for bounded intervals $D$ and fixed $c$.

Finally similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous time models including branching Brownian motion.

www.shef.ac.uk /~st1jdb/mart-www.html (192 words)

PlanetMath: compact-open topology(Site not responding. Last check: 2007-10-07)

The corresponding question for uniformlimit spaces has been solved by Wyler [19], whereas completions in the realm of filter spaces have been constructed by Kent and Rath [11] as well as by Császár [6].

The category Conv of convergence spaces (and uniformly continuous maps) is isomorphic to the category KConvs of symmetric Kent convergence spaces (and continuous maps); it is a bicoreflective subcategory of SUConv (cf.

[No title](Site not responding. Last check: 2007-10-07)

Since the text book includes some more difficult results about the topic of uniformconvergence, and which we do not expect you to know yet, let me state explicitly what we DO expect you to know at this stage about uniformconvergence of Fourier series.

You should already know (also from earlier courses) the definition of uniformconvergence of a given series of functions on a given interval.

The Galerkin convergence curves are the same ones that were presented in Figure 6.5, and are included here for comparison with GLS convergence.

We consider first the GLS convergence with uniform refinement, and hereafter refer to it as GLS/uniform (analogous notation is used for the other cases).

Figure 6.8 repeats the convergence study of Figure 6.7, this time showing the estimated error as a function of mesh refinement.

Abstract of: Uniform convergence of curve estimators for ergodic diffusion processes(Site not responding. Last check: 2007-10-07)

Using empirical process theory for martingales, we first prove a theorem regarding the uniform weak convergence of the empirical density.

This result is then used to deriveuniform weak convergence for the kernel estimator of the invariant density.

For kernel estimators of the derivatives of the invariant density and for a nonparametric drift estimator that was proposed by Banon, we give bounds for the rate at which the uniform distance between the estimator and the true curve vanishes.

NeuroCOLT: Neural Networks and Computational Learning Theory(Site not responding. Last check: 2007-10-07)

Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers.

These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables.

In this paper we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes.

Gerhard Preuss(Site not responding. Last check: 2007-10-07)

Recently semiuniform convergence spaces have been studied by the author as a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniformlimit spaces (and thus of uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients.

They form the suitable framework for studying continuity, Cauchy continuity and uniformcontinuity as well as convergence structures in function spaces, namely simple convergence, continuousconvergence and uniformconvergence.

Furthermore, the localization of compactness (when considered in the realm of semiuniform convergence spaces) leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the locally compact spaces are exactly the compactly generated spaces (even without the Hausdorff axiom!).

To cover basic ideas of convergence and completeness in R. To introduce the idea of uniformconvergence.

Sequences and series, uniformconvergence, the Riemann integral, convergence of Fourier series, applications, functions and distributions, transform techniques, existence and uniqueness for PDEs, fundamental solutions and Green's functions.

Uniformconvergence [5] Definition and illustrations; uniformlimit of continuous is continuous; uniformconvergent series and the M-test; term-by-term integration and differentiation; examples and applications.