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 |