Class Diary for M447, Fall 2015, Jochen Denzler
Wed Aug 19:
Notation; sets, and basic logic associated with them.
Fri Aug 21:
Functions; images and inverse images; injective, surjective, bijective.
Field axioms (as part of the axioms for the real numbers).
Mon Aug 24:
Hwk discussion; Ordered Fields; examples.
Wed Aug 26:
Hwk discussion; order on R(X); Archimedean property; R(X) is an example of a
non-archimedean ordered field;
Fri Aug 28:
The supremum axiom; The reals defined as a (the) ordered field satisfying the
supremum axiom; brief outlook on construction of integers and rationals from
natural numbers (will comment on construction of reals a bit
later). Archimedean property proved from sup axiom
Mon Aug 31:
Dedekind cuts quickly for info purposes;
More on Archimedean property; construction of complex numbers from reals;
Limits of sequences.
Hwk 1-3 due Friday (some info on #3 will come on
Wed yet)
Wed Sep 02:
Limit laws. limsup and liminf.
Fri Sep 04:
Proofs with lim / limsup / liminf. Cauchy sequences and their convergence.
Hwk 4-8 due Friday.
Mon Sep 07:
LABOR DAY
Wed Sep 09:
Normed vector spaces; definition and examples; convergent sequences
and Cauchy-sequences.
Fri Sep 11:
Metric spaces, convergent and Cauchy sequences therein. Completeness.
Mon Sep 14:
Banach's fixed point theorem.
Hwk 9-14 due Friday if possible.
Wed Sep 16:
Some applications of BFPT mentioned.
Continuity and uniform continuity for mappings btw metric spaces.
Fri Sep 18:
Continuity of exp function proved; pointwise and uniform convergence of
functions defined.
Hwk 15-18 due Friday if possible.
Mon Sep 21:
Proof for uniform limits of continuous functions; Hwk 7 returned and amply
discussed. Hwk 15-20 due Monday (moving deadline
from Friday and updating file)
Wed Sep 23:
Hwk 7 a few more comments; Bolzano Weierstrass;
Fri Sep 25:
continuous fcts on closed bdd interval; sequential compactness;
Intermediate value theorem; balls in metric spaces.
Mon Sep 28:
Open and closed sets; topology. new hwk 21-26, 29
due Monday (27,28 have more time)
Wed Sep 30:
boundary; continuity in terms of open sets; some counterexamples for
f(open)=open and the like
Fri Oct 02:
a counterex. involving the Cantor set; connectedness and examples
Mon Oct 05:
connected sets in the reals; continuous fcts map connected into connected
Hwk 27,28, 30-32 are hereby assigned, but will not be collected. Sols posted
already
Wed Oct 07:
EXAM 1
Fri Oct 09:
Cover compactness, and an example thm (uniform continuity). We will see it is
equivalent to sequential compactness.
Mon Oct 12:
Totally bounded (and motivation of the notion. Equivalence of compactness
proved (almost finished)
Wed Oct 14:
Equivalence proof finished; consequences.
Fri Oct 16:
FALL BREAK
Mon Oct 19:
exam return and discussion; a bit more on compactness (pg 59 of handwritten notes)
Wed Oct 21:
Arzela Ascoli started; motivation, outline, diagonal sequences
Fri Oct 23:
A-A finished; converse started.
Mon Oct 26:
A-A converse finished. Equivalence of norms in R^n
Wed Oct 28:
Derivatives in single variable
Fri Oct 30:
derivative of inverse function; spaces of differentiable functions
Mon Nov 02:
Metric, not norm, for C^infinity. Smooth cutoff fct.
Mean value theorem and consequences
Wed Nov 04:
Lipschitz continuity; uniqueness for some diffeqs.
Fri Nov 06:
Uniqueness for some Diff-Eq's. Taylor's formula
Mon Nov 09:
Series; Power series; convergence tests.
Wed Nov 11:
the exponential series; identified with exp fct by E'=E, then extended to
complex arguments; functional equation of exp from power series
Fri Nov 13:
Cauchy product; convergence pf for Cauchy product of two abs convergent series.
Mon Nov 16:
power series of sin and cos defined; easy consequences
Wed Nov 18:
EXAM 2
Fri Nov 20:
leftovers:
Cauchy-Hadamard formula for power series; addition thm for sin, cos via
Taylor's formula; analytic functions defined;
Mon Nov 23:
power series of inverse function: convergence proof by majorant method
Wed Nov 25:
Fri Nov 27:
THANKSGIVING BREAK
Mon Nov 30:
Wed Dec 02: STUDY DAY
Mon Dec 07: FINAL EXAM 08:00-10:00
(scheduled by university policy), see
exam schedule by class.
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