Class Diary for M447, Fall 2009, Jochen Denzler
Wed Aug 19:
Quick review of logic through p 19. Hwk: Ex 2.4 (not collected)
Fri Aug 21:
Functions; injective; surjective etc. Field axioms Hwk.
from 2.20 the part relating to (2.5); 2.21.2; 2.27
Mon Aug 24:
Ordered fields; the supremum axiom; The example R(X) as an ordered field
that doesn't satisfy the sup axiom (and not the archimedean property either)
Wed Aug 26:
Properties and proofs for supremum. Hwk: 2.32, 2.34.2 due Fri
Fri Aug 28:
Supremum finished up; Archimedean property as a consequence of sup axiom.
A 10 min overview how one could (alternatively) construct the reals from scratch.
Mon Aug 31:
Hwk comments. Cardinality. A few paradoxes mentioned.
Wed Sep 02:
Sequences. Metric spaces barely defined.
Fri Sep 04:
Examples for metric spaces, and basic consequences of the definition.
Hwk: draw balls for 1,2,and infinity distances respectively in R^2,
possibly also for a few other p if you wish
-- see addenda
Mon Sep 07:
LABOR DAY
Wed Sep 09:
another example of a metric space: rationals with p-adic distance.
-- Open and closed sets. Hwk: Show that A-interior is always open,
A-closure is always closed, and that the complement of the closure of a set is
the interior of thhe complement. Not for turn-in -- Here is
your Hwk for grading due Mon
Fri Sep 11:
Sequences in metric spaces; limit and cluster points.
Mon Sep 14: Continuity. Hwk due Friday for
grading. Also read sec 3.4 carefully by Wed
Wed Sep 16:
Comments on returned hwk; including well-ordering principle (see sample sols
made availbale). Continuity finished up. limsup and semi-continuity.
Fri Sep 18:
comments on pending hwk; New hwk on semicontinuity Exercise 3.4.X1, (1)-(4)
due Wed for grading. Sequential compactness. [For later convenience,
please amend all references to compactness in Sec 3.5 before Rmk 3.5.13. Write
`sequentially compact' instead of compact.]
Mon Sep 21:
Heine-Borel. Continuity and compactness. Cover compactness defined and
motivated.
Wed Sep 23:
Discussion of returned homework. Handed out notes on additional material.
Hwk deadline for 3.4.X1 extended
Fri Sep 25:
Basic theorems about cover compactness
Mon Sep 28:
compactness continued
Wed Sep 30:
equivalence of cover and sequential compactness
Fri Oct 02:
compactness finished up.
Mon Oct 05:
Cantor set and fat Cantor sets; subspace matric: open and closed as relative
notions.
Wed Oct 07:
subspace metric: continuity, compactness, isometries, homeomorphisms
Fri Oct 09:
product metrics; topological equivalence.
Mon Oct 12:
EXAM 1
Wed Oct 14:
Lipschitz functions. Product of compact metric spaces is compact. Rn
begun; vector space operations; norm, inner product, Cauchy-Schwarz.
Fri Oct 16:
FALL BREAK
Mon Oct 19:
Product spaces finished, and Heine-Borel for R^n
Wed Oct 21:
Connectedness.
Fri Oct 23:
Connectedness finished; metric completeness. Hwk: read and hand in problem
solutions on the first page of this by Wed
Mon Oct 26:
Comments about Fri hwk; Complex numbers
Wed Oct 28:
comments on belated semicontinuity, and on new hwk. complex sequences.
Fri Oct 30:
More on sequences; Inequality of arithmetic and geometric mean proved and used
for an alternative proof for Ex 4.12
Mon Nov 02:
Series. Hwk till next Mon: from addenda, Hwk3.11.X1 and X2; read but don't
do X3. Also do Ex 4.22 from notes; but I won't collect it for grading.
Wed Nov 04:
proof of multiplicatioon of series theorem.
Fri Nov 06:
alternating series; an example of two conditionally convergent series whose
(formal) product is divergent; rearrangement of absolutely convergent series;
all the weird stuff caused by rearrangement of conditionally convergent series
Mon Nov 09:
Cauchy test for series with decreasing, positive terms. Feedback for hwk
3.9X1-3: uniform convergence vs pointwise convergence.
--- Tue Nov 10: WP/WF drop deadine ---
Wed Nov 11:
formal power series; convergent power series; radius of convergence
Fri Nov 13:
reciprocal of a power series with non-zero constant term; formal and
convergence proof by majorant method. (See notes handed out)
Mon Nov 16:
pointwise and monotone convergence; uniform convgence
Wed Nov 18:
uniform convergence, exchange of limits, and consequences
Fri Nov 20:
Uniform convergence of power series on compact disks inside the disk of
convergence. Differentiation of power series.
Mon Nov 23:
Wed Nov 25:
EXAM 2
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)
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