Uniform convergence on a set VS Uniform convergence on all of its compact subsets

Question

Let {$f_n$}$_{n=1}^{\infty} $ be a sequence of functions defined on a subset $A$ of the complex plane(Or the real line). What's the difference between saying that {$f_n$}$_{n=1}^{\infty} $ is uniformly convergent on $A$ and saying that {$f_n$}$_{n=1}^{\infty} $ is uniformly convergent of all compact subsets of $A$? I understand what both statements mean but I want to know what different implications each statement has. Thanks.

Answer 1

$f_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$ is uniformly convergent on every compact subset of $\Bbb C$, but not on $\Bbb C$ as a whole.

Thus uniform convergence on all compact subsets is the more flexible and useful variant.

Answer 2

An example from complex analysis: Let $D$ be a non-empty open subset of $\mathbb C, $ and let $(f_n)_n$ be a sequence of functions that are analytic on $D.$ If $(f_n)_n$ converges uniformly to $f$ on every compact subset of $D$ then $f$ is analytic on $D,$ and $(f'_n)_n$ converges on $D$ to $f'.$

A corollary is that the sum of a power series is analytic inside its circle of convergence, and within this circle it may be differentiated term-by-term to get the derivative of the sum of the series. But the series may fail to converge uniformly on all of $D.$ For example, $D=\{z:|z|<1\}$ and $f_n(z)=\sum_{j=0}^nz^j.$

Answer 3

Uniform convergence on compact subsets is clearly a more general phenomenon than uniform convergence on the entire plane (as LutzL's example shows).

In the context of complex analysis, uniform convergence of $(f_n)$ on compact subsets is good enough to conclude that the limit function $f = \lim_n f_n$ is holomorphic. The easy proof of this is to use Morera's criterion for holomorphy and take the limit inside the integral over triangles. This is the most likely scenario where uniform convergence of any sort is important. There is also a more general version of this where we replace holomorphic functions by meromorphic functions and the complex plane by the Riemann sphere.

There is a body of important results relating to when you can extract from a family a sequence of functions that is uniformly convergent on compact sets: see the Wikipedia article for a normal family. These are closely related to the Ascoli-Arzela theorem from real analysis. Among other things, they can be used to prove the Riemann mapping theorem, which is up there among the more interesting and unexpected results from complex analysis.

In greater generality, we can also speak of abstract topological spaces and uniform convergence on compact subsets or uniform convergence on the entire space. Let me call them UC (uniform convergence) and UCCS (uniform convergence on compact subsets) for short. These notions will coincide if the base space is compact: for example, in the space $C([0,1])$ endowed with the supremum norm, UC and UCCS are equivalent. However, when the base space is not compact there is a difference: again, LutzL's example is a nice illustration here.

UCCS is typically used as a replacement for UC on noncompact spaces. The main useful feature of uniform convergence is that it preserves certain properties of the sequence in the limit: the well-known ones are continuity, integrability (on a compact space), and (with additional assumptions) additional regularity, like differentiability or Holder continuity. Some of these interesting properties, however, are purely local: for example, continuity and differentiability are properties of functions that depend only on nearby values of the function. To preserve such a local property, it makes sense that we only need uniform convergence in a local sense as well, which is the role that UCCS plays. This sort of thing becomes more useful as you develop more tools in functional analysis.