Week

Monday

Tuesday

Thursday

Sept 4-7

Labor Day

First day of class:

Metric Spaces, Normed Vector Spaces

problem for Thursday

Ex of norms, Cauchy Schwarz inequality, Neighborhood

Page 42 #17

Sept 10-14

Convergence in Metric Spaces, Sequence spaces

page 47 #39

The space C([a,b]) with the sup-norm and the L1-norm, open sets, closed sets.

Closed sets, interior and closure, limit points

page 58 #34

Sept 17-21

Ā=AUA', exemples in l2

practice pb from class

l2 cont'd, ex in C([0,1]), boundary points

class pb given in class

Continuity for fonctions between Metric spaces, ex, relative topology

page 65 #8

Sept 24-28

Continuity cont'd, homeomorphism, isometry, Space of Continuous fcts. Completeness

Completeness of the k-dim Euclidean space, of lp spaces; Sequentially Compact Space

Test 1

Oct 1-5

Seq Compact space cont'd, compact sets (covering), compact is sequentially compact

Seq compact is compact (via Totally Boundedness). f(K) is compact, f achieves its max and min on K, unif cont.

Cont functions on compact sets are unif cont, pointwise and uniform convergence, the space of Continuous functions between metric spaces.

Oct 8-12

Fall Break

Fall Break

Fall Break

Oct 15-19

unif limits of continuous functions is continuous, unif metric is the good metric for C(M,N) when M is compact or for C_b(M,N); the Space is complete if N is complete; interchange of limits and integrals under unif conv, ex

p149, #9c,d

Ex of unif convergence of fcts, review of properties of R. I., more ex of interchange with int, interchange of limit with derivative

p.153#19

proof of the thm on interchange of limit and derivatives, unif conv of series of functions, Weierstrass M test, corollaries to interchange of limits and integrals and derivatives for series of continuous functions, a cont function that is nowhere differentiable.

Oct 22-26

continuous but nowhere differentiable function, Weierstrass Approximation Theorem

Proof of the Weierstrass Approximation theorem using the Approximation to the Identity approach

prac pb: if all moments of a funtion are zero then the fct is zero

Application of the Weierstrass Approximation Thm, applications of the Approximation of the Identity

Oct 29-Nov 2

examples from the practice pb

Test 2

Compact set in the space of continuous functions, the Arzela-Ascoli thm

Nov 5-9

The equivalence between compact sets and equicontinuity, Euler's approximation and Peano's thm

End of the proof of Euler's approximation for ODE's

Fourier series: an introduction. The norm L^2, the n-th Fourier sum is the best L^2 approximation in the linear supbspace generated by 1, cos(kx) and sin(kx), k=1,..,n

Nov 12-16

Bessel's inequality and Parseval's identity, Fourier coefficients in l1 implies unif conv

Complex form of the Fourier series, derivation of the Dirichlet kernel, Properties of the Dirichlet kernel

pbs with pointwise convergence, Dini's criterion, Cesaro summability, Fejér's kernel

Nov 19-23

Uniform Cesaro summability of the Fourier Series to periodic continuous functions, and hence L2 convergence of the Fourier series. Corollary: uniform conv of the Fourier Series to periodic, C1 functions. Intro to Lebesgue Theory

Intro to Lebesgue Theory cont'd: sigma-algebra, measurable space, measures, ex

Test 3

Nov 26-30

Lebesgue measure, properties, notion of a.e., measurable fcts

measurable functions form an algebra, max, limits of measurable functions are measurable, simple functions, approximation of non-neg measurable functions by a monotone sequence of simple functions

approx of measurable functions by simple functions, def of the Lebesgue integral of non-neg simple functions, of measurable functions, properties of the integrals, Monotone Convergence Thm

Dec 3-5

Proof of the MCT, Fatou's Lemma, linearity of the Lebesgue integral, Lebesgue Dominated Convergence Thm

Proof of the LDCT, (proper) Riemann integral is L^1 with same value.

LAST DAY