Basics

Compactness

theorem (Folland 0.25):

For E(X,d) a metric space, TFAE:

  • E is complete and totally bounded.
  • E is sequentially compact: Every sequence in E has a subsequence that converges to a point in E.
  • E is compact: every open cover has a finite subcover.

Note that E is complete as a metric space with the induced metric iff E is closed in X, and E is bounded iff it is totally bounded.

Topology / Sets

proposition (Compact if and only if sequentially compact for metric spaces):

Metric spaces are compact iff they are sequentially compact, (i.e. every sequence has a convergent subsequence).

#todo Proof

proposition (A unit ball that is not compact):

The unit ball in C([0,1]) with the sup norm is not compact.

proof (?):

Take fk(x)=xn, which converges to χ(x=1). The limit is not continuous, so no subsequence can converge.

theorem (Heine-Borel):

XRn is compact X is closed and bounded.

proposition (Geometric Series):

k=0xk=11x|x|<1.

corollary (?):

k=012k=1.

proposition (?):

Singleton sets in R are closed, and thus Q is an Fσ set.

lemma (?):

Any nonempty set which is bounded from above (resp. below) has a well-defined supremum (resp. infimum).

Smallness for sets

proposition (Finite unions of nowhere dense sets are still nowhere dense):

A finite union of nowhere dense is again nowhere dense.

theorem (Baire):

R is a Baire space, i.e. R can not be written as a countable union of nowhere dense sets.

proposition (?):

The Cantor set is closed with empty interior.

proof (?):

Its complement is a union of open intervals, and can’t contain an interval since intervals have positive measure and m(Cn) tends to zero.

corollary (?):

The Cantor set is nowhere dense.

Smallness for functions

proposition (Existence of Smooth Compactly Supported Functions):

There exist smooth compactly supported functions, e.g. take f(x)=e1x2χ(0,)(x).

  • Arzela - Ascoli 1: If F is pointwise bounded and equicontinuous, then F is totally bounded in the uniform metric and its closure ¯FC(X) in the space of continuous functions is compact.

  • Arzela - Ascoli 2: If {fk} is pointwise bounded and equicontinuous, then there exists a continuous f such that fkuf on every compact set.

#todo Proof

  • Bolzano-Weierstrass: Every bounded sequence has a convergent subsequence.

  • Heine-Borel: XRn is compact X is closed and bounded.

  • Baire Category Theorem: If X is a complete metric space, then X is a Baire space:

    • For any sequence {Uk} of open, dense sets, kUk is also dense.
    • X is not a countable union of nowhere-dense sets
  • Nested Interval Characterization of Completeness: R being complete for any sequence of intervals {In} such that In+1In, In.

  • Convergence Characterization of Completeness: R being complete is equivalent to “absolutely convergent implies convergent” for sums of real numbers.

  • Compacts subsets KRn are also sequentially compact, i.e. every sequence in K has a convergent subsequence.

  • Closed subsets of compact sets are compact.

  • Every compact subset of a Hausdorff space is closed

  • Urysohn’s Lemma: For any two sets A,B in a metric space or compact Hausdorff space X, there is a function f:XI such that f(A)=0 and f(B)=1.

  • Continuous compactly supported functions are

    • Bounded almost everywhere

    • Uniformly bounded

    • Uniformly continuous

      Proof:

      figures/2019-12-19-16-49-56.png

  • Uniform convergence allows commuting sums with integrals

#todo