Professor: Thea Pignataro
Read Ross secs 13, 21 & 22.
Review linear algebra: Subspace, spanning set, linear independence, dimension, basis, kernel, range, std. basis, affine subspace.
Prof's notes on http://1drv.ms/1KlFihR, mirror: link1
Sec. 1.6
Def: a normed real vect. space. (Theorem 1.6.6)
Generalization of vector length
Inner product space (theorem 1.6.6 I)
Remark: Every inner product induces a norm, but not every norm can be represented as an inner product. Nor does every metric arises from a norm.
Sec. 1.7 Matrices, etc.
Def. 1. 7.1 A metric space.
Check example 1.7.2 (discrete methods), many set, d(x, y) = 0 if x=y and d(x,y) = 1 if x != y. Check that this is a metric.
02/03/2016
Ch. 2 Topology of Metric Spaces
Open sets
Def. 2.1.1
Prop. 2.1.3 (i easy, ii proven, iii trivial)
Interior 2.2
Def. 2.2.1
Closed 2.3
Def.2.3.1
02/08/2016
HW #01
1.6 #1,3,4;
1.7 #2,3;
Ch 1 #10,12, read 14&20
2.1 #1,3;
2.2 #1;
2.3 #1,2,5
2.4 #4
2.5 #1,3,4
2.6 #2,3
2.7 #1
2.8 #2,3
2.9 #3
Ch 2 #44(read) 19, 22
2.4 Accumulation Points (pg. 127)
Theorem (2.4.2)
2.5 Closure
Definition 2.5.1
Remark: So cl(A) is the smallest closed set containing A.
2.6 Boundary
pts of A & pts of M\A =
= pts. not in A arbit. close to any boundary pt.
Def. 2.6.1
ie
remark: bd(A) is closed
Examples given
2.7 Sequences
Definition: 2.7.1
Prop. 2.7.2 (pg. 134)
, then any open set containing x contains all but finitely many 
.
. Need to deal with bdd.
Remark: 1.
2. a lot of seq. then carry over to metric or normed spaces (e.g. prop. 2.7.3) but there is no longer a notion of order. Therefore, no notion of monotonic (unless we are talking about R) or sup/inf of subsets of M (only of sets of distances
3. Limits in metric spaces are unique (pf. just like for R since triangle ineq. holds for d).
Prop. 2.7.4
Proof. n finite is crucial
outline of pf. in book.
So take N =
02/10/2016
Prop. 2.7.6, with proofs.
Prop. 2.8 Completeness (check all), all of 2.9
Chapter 3. Compact & Connected Sets
02/17/2016
HW for Mon: 3.1 #4,5
02/24/2016
of sets is nested iff 
.
3.3 Nested Sets
Def. a seq.
Theorem (3.3.1) If F_k is a nested seq. of compact, non-empty, sets in a metric sp(M,d), then
Pf: Method 1, p.157, method 2, p.169
Theorem: Let C be a collection of closed sets in
with the finite 
prop. s.t. 
an element of C which is bdd, 
.
}, 
closed 
s.t. 
bdd.
so 
. Suppose not, ie. 
.
is in each 
= open cover of 
Pf: let C = {
&
s.t.
So no pt. of
So
ie.
Since
s.t. 
is open and contains y.
closed & bdd => 
compact => 
finite sub-cover 
.
prop.
but then
# C has finite
3.5 Connected Sets (some lectures found on youtube)
, then 
or 
.
When is a set in "one piece"?
Def: A set S in a space X is connected IFF whenever
Two open sets U, V in X are said to separate S IFF
i).
ii).
iii).
iv).
If
such U & V, then we call S disconnected.
So, 1st definition is negation of disconnected.
Pf:
. Then 
, open
, open
So,
And
and
and
=> exercise where you consider
03/09/2016
Quiz #1 on chapter 1, 2, 3 on Wed 3/16 at start of class.
Quiz #1. hi: 49.5, low: 30, mean: 42.8, median: 45.5
05/02/2016
Catch up:
Jacobian
LaGrange
Lagrange Multiplier (not covered, but read for Math major's sake) also in multivar calc. textbook.
05/09/2016
last day Wed 5/18, Quiz 2:
No proofs for 7.1, 7.2.
Final: NAC 4/115, 3:30pm - 5:45pm Monday 5/23
05/18/2016
Final:
Expect basic proofs.
Office Hours Th 5/19, 2-4pm, NAC 6/202B
Review Class NAC 1/511E
(Artino Classroom) 6pm ->? Pizza
Final Exam Mon 5/23, 3:30PM _ 5:45PM in NAC 4/115
Theorem list for knowing proofs from "midterm".
Definition-type proofs (proofs by def)