\chapter{Uniformity}
\epsilon-\delta says at each point, do this;
uniformity say across some family (of points, or of functions)
\begin{ped}
These are all examples of
"The problem of a single example":
each concept is introduced to prove a single theorem:
- uniform limit of continuous is continuous
- uniformly continuous is integrable
- equicontinuous, totally bounded family is compact
They can be made richer;
the uniform *metric* is a good EG of a metric,
of a topology on function spaces,
and $L^\infty$ is a nice space
\end{ped}
\section{Definitions}
uniform convergence
uniform continuity
$$\forall epsilon \exists delta s.t. \forall x$$
uniform across *points* (for given epsilon)
equicontinuity
$$\forall x, epsilon \exists delta s.t. \forall f$$
uniform across *functions* (at a given point, for given epsilon)
uniform equicontinuity
$$\forall epsilon \exists delta s.t. \forall f, x$$
``uniformly uniformly continuous"
\section{Uniform continuity}
Easiest eg is bounded derivative
Lipschitz
[bounded derivative -> Lipschitz with exponent 1, by MVT]
note re: uniform continuity (inc. Lipschitz):
at any given scale, stretches by a controlled amount
...but can stretch infinitely infinitesimally
(EG, $x^{1/3}$)
Indeed, does so for exponent < 1.
(Lipschitz with exponent 1 (inc. bounded derivative)
doesn't stretch infinitely infinitesimally)
Main reason we care in calc is:
\begin{thm}
Uniformly continuous (on compact set) is Riemann integrable.
\end{thm}
\begin{proof}
Easy: given \epsilon (for integrals), choose partition smaller than
``a $\delta$ that works for $\epsilon' = \epsilon/\mu(A)$''
Then on each piece, sup and inf are within \epsilon' of each other,
so lower and upper sums are within \epsilon of each other
\end{proof}
Key application:
\begin{thm}
Continuous functions on a compact metric space are uniformly continuous.
[or at least the interval?]
\end{thm}
\begin{cor}
Continous functions on compact XXX (interval, etc.) are Riemann integrable.
\end{cor}
Lipschitz is used in coarse geometry;
Uniform continuity in constructible calc?
\section{Equicontinuity}
ArzelĂ -Ascoli:
This is about compact sets in function spaces
(like Heine-Borel)
A uniformly bounded, equicontinuous family of functions
on a compact space (or just interval?)
has a subsequence that converges uniformly.
IE:
on C(K) [continuous on compact]
- uniformly bounded
- equicontinuous
=> compact (in uniform metric)
[Is this "iff"?
compact -> uniformly bounded (else take "max")]
EG if f & f' are uniformly bounded
General form:
Let X be a compact metric space, Y a metric space. Then a subset F of C(X,Y) is compact if and only if it is equicontinuous, pointwise relatively compact and closed.
(cf. Montel's theorem for holomorphic analog)
equicontinuous:
pointwise limit (on a dense subset) is continuous
(hmm...b/c it implies compact-open convergence,
hence on locally compact space)
on compact: pointwise convergence -> uniform convergence
\section{My notes}
This section doesn't offer much insight;
it's just a listing.