Syllabus
Syllabus
Wk Lec Date Topic Royden Hwk
1


2


3


4


5


6


7


8


9


10


1
2
3
4
5
6

7
8
9
10
11
12
13
14
15
16
17

18
19
20
21
22
23
24
25
26
27
28
01/07
01/08
01/09
01/14
01/15
01/16
01/21
01/22
01/23
01/28
01/29
01/30
02/04
02/05
02/06
02/11
02/12
02/13
02/18
02/19
02/20
02/25
02/26
02/27
03/04
03/05
03/06
03/11
03/12
03/13
Deficiencies of Riemann integral.
Open and closed sets.
Sigma-algebras. Outer measure and measurability.
Measurable functions. The Lebesgue integral on a line.
Convergence theorems. Convergence in measure.
The Lebesgue integral in n dimensions. Fubini and Tonelli theorems.
NO CLASS. Martin Luther King Day holiday.
Differentiation and integration.
Absolute continuity. Change of measure. Radon-Nikodim theorem.
Lp-spaces.
Norm. Completeness. Linear functionals in Lp-spaces.
Banach spaces. Examples.
Linear operators. Linear functionals
Norm of an operator.
Hahn-Banach theorem.
Open mapping theorem.
Closed graph theorem.
Duality. Weak topologies.
NO CLASS. President's Day holiday.
Elliptic boundary value problem.
Mollifiers.
Distributions.
Examples of distributions.
Sobolev Spaces.
Sobolev embedding theorem.
Trace
Kernel, range of an operator.
Integral operators. Compact operators.
Fredholm alternative.
Review.
4.1,
2.5, 7.2, 8.1
3.2-4, 11.1, 12.1-2
3.5-6, 11.2
4.2-5, 11.4
12.4

5.1-5.4
11.6
6.1-2
6.3-4
10.1
10.2
10.3
10.4
10.6
10.6
10.8

R. Showalter's book Ch. 3.
R. Showalter's book. Ch. 2
R. Showalter's book. Ch. 2
R. Showalter's book. Ch. 2
R. Showalter's book. Ch. 2
R. Showalter's book. Ch. 2
R. Showalter's book. Ch. 2









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