Re: Why do people make such big deals of uniform convergence/continuity?

In article <1158667818.342560.163650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Doug B <doug_protocols@xxxxxxxxx> wrote:
Why do people always make such huge deals of uniform convergence and
uniform continuity? These non topological notions seem to get a
disproportionate amount of attention vs. how useful they are.

And yes, I know someone is going to say: "integral of a limit...
derivative of a series..." but that just transforms the question to,
why do people make such big deals about integrals of limits /
derivatives of series? Integrals of limits are important, but in true
real analysis these are almost always studied using dominated
convergence, which is how God meant us to study them, not via the
clunky unwieldy abomination of uniform convergence.

I'm not trying to just whine, I seriously am interested-- are there
really areas in higher math, areas that are actively researched and
which spark interest, where anything remotely important hangs in the
balance of whether some sequence of functions converges uniformly?

In complex analysis, for example, the standard mode of convergence
for a sequence of analytic functions is uniformly on compact sets,
from which all sorts of good things flow.

One well-known consequence of a lack of uniform convergence is the
Gibbs phenomenon in Fourier series. I'm told this has quite practical
effects in electrical engineering.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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