# Math 447, Principles of Analysis II, Spring 2009

Harold P.
Boas

- Monday, May 11
- The final exam was given, and solutions are available.
- Monday, May 4
- In our last class meeting for the semester, we completed the proof of the Riesz representation theorem characterizing the dual space of C[0,1].
- Wednesday, April 29
- We worked on the proof of the Riesz representation theorem. We found what the representing function α has to be, and we checked that α has bounded variation. The remaining step is to show that α does represent the given linear functional L.
- Monday, April 27
- We discussed the general notion of the dual space of a normed space and then specialized to the setting of the Riesz representation theorem characterizing the dual of C[0,1]. In preparation for the proof, we observed that a function α of bounded variation representing (via Stieltjes integration) a given linear functional on C[0,1] is not unique, but can be made unique by insisting that α be continuous from the right and that α(0)=0.
- Wednesday, April 22
- We compiled a list of properties of the Stieltjes integral, defined the Stieltjes integral for integrators α of bounded variation (not just monotonic α), and discussed the implications of the inequality stating that |∫
_{a}^{b}*f*dα| ≤ ||*f*||_{∞}V_{a}^{b}(α), which partly motivates the Riesz representation theorem.

Solutions to the assignment due today are posted at the TAMU eLearning site. - Monday, April 20
- We looked at some more examples of functions that are or are not Riemann–Stieltjes integrable, and we discussed Riemann's criterion for integrability. We stated the Riesz representation theorem about the dual space of C[0,1]; the theorem that ∫
_{a}^{b}*f*dα exists for every continuous function*f*and every increasing function α; and the theorem that ∫_{a}^{b}*f*(x) dα(x)=∫_{a}^{b}*f*(x)α'(x) dx if*f*is continuous and α is continuously differentiable. As an aside, we discussed the Cantor function, which is not equal to the integral of its derivative. - Wednesday, April 15
- The next (and last) assignment is posted. Solutions to today's assignment are posted at the TAMU eLearning site.

In class, we defined the Riemann–Stieltjes integral and worked out an example with a step function. - Monday, April 13
- We continued the discussion of functions of bounded variation, observing that the space BV[a,b] is a complete normed space. Then we studied the sequence of functions x
^{n}in the normed spaces B[0,1], C[0,1], BV[0,1], L_{1}[0,1], and L_{∞}[0,1]. Also we did the SLATE evaluation forms.

Next time we will start discussing the Stieltjes integral. - Wednesday, April 8
- We stated the Dirichlet–Jordan theorem about pointwise convergence of Fourier series of functions of bounded variation, introduced a norm on the space of functions of bounded variation, discussed convergence with respect to this norm, and proved Jordan's decomposition theorem for functions of bounded variation.

The next assignment is posted. - Monday, April 6
- We discussed the definition and some examples of functions of bounded variation, a notion that Camille Jordan introduced [Sur la série de Fourier, C. R. Acad. Sci. Paris 92 (1881) 228–230] in his generalization of Dirichlet's work on Fourier series. We stated Jordan's result that every function of bounded variation can be expressed as the difference of two monotonic functions.
- Wednesday, April 1
- The next assignment is available in pdf.

In class, we proved two theorems about convergence of Fourier series: (1) if an integrable function has a derivative at some point, then the Fourier series of the function converges at that point (to the value of the function); (2) the Cesàro averages of the partial sums of the Fourier series of an L_{1}function converge to the function in L_{1}norm. The proofs use the Dirichlet and Fejér kernels. - Monday, March 30
- We proved Parseval's equation (equality in Bessel's inequality), discussed the isometry between L
_{2}[-π,π] and ℓ_{2}, and introduced the Dirichlet kernel and the Fejér kernel. - Wednesday, March 25
- We discussed the Riemann–Lebesgue lemma and Bessel's inequality.

Solutions to today's assignment are available at the TAMU eLearning site, and the next assignment is posted. - Monday, March 23
- We began the discussion of Fourier series and stated some milestones in the subject: du Bois-Reymond's example of a continuous function whose Fourier series has a point of divergence; the corollary via the Baire category theorem that there exists a continuous function whose Fourier series diverges on a dense set; Kolmogorov's example of a Lebesgue-integrable function whose Fourier series diverges everywhere; and Carleson's theorem that the Fourier series of a square-integrable function converges almost everywhere.

The assignment for next class is to do exercise 7 on page 250 (in Chapter 15). - Wednesday, March 11
- The midterm exam was given, and solutions are available.

There is no assignment over spring break. - Monday, March 9
- We discussed Vitali's construction (using the axiom of choice) of a nonmeasurable set.
- Wednesday, March 4
- We reviewed for the midterm exam to be given on Wednesday, March 11 on Chapters 16, 17, and 18.

Solutions to the assignment due today are available at the TAMU eLearning site. - Monday, March 2
- We proved that the normed space L
_{1}is complete. - Wednesday, February 25
- The next assignment is
posted.

In class we discussed the applicability of convergence theorems to some concrete examples of integrals, and we proved Fatou's lemma and Lebesgue's dominated convergence theorem. - Monday, February 23
- We discussed the definition of the Lebesgue integral for general measurable functions, and we made a list of six convergence theorems.
- Wednesday, February 18
- The next assignment is
posted.

In class, we defined the Lebesgue integral of a nonnegative function, and we stated the monotone convergence theorem. - Monday, February 16
- We discussed Littlewood's three principles, the proof of Egorov's theorem, and the notion of measurability of functions taking values in the extended real numbers.
- Wednesday, February 11
- The next assignment is
posted. Solutions to the first three assignments are available at the TAMU eLearning site.

In class, we discussed the properties of the set of all measurable functions: it is a vector space, an algebra, a lattice, closed under modifications on sets of measure zero, closed under taking pointwise limits, and generated by continuous functions (in the sense that every measurable function is the pointwise limit almost everywhere of a sequence of continuous functions). - Monday, February 9
- We discussed the theorems of Borel and Luzin stating that a measurable function is "nearly" continuous, and we proved that a measurable function is the pointwise limit of an increasing sequence of simple functions.
- Wednesday, February 4
- We finished the proof from last time, using the idea that a measurable set can be approximated by a union of intervals. Then we returned to the topic of measurable functions.

The next assignment is posted. - Monday, February 2
- We discussed the σ-algebras of Borel sets and of Lebesgue-measurable sets and the principle that a measurable subset (of the real line) having finite measure is "nearly" a finite union of open intervals. Then we started to prove for a measurable set E of finite measure that the measure of E∩(E+x) is a continuous function of x, but we ran out of time.
- Wednesday, January 28
- We discussed measurable sets (with an aside on nonmeasurable
sets, the axiom of choice, and the Banach–Tarski paradox). Also we
had a preview of the topic of measurable functions, the subject of the next chapter.

The next assignment is posted. Solutions to the assignment due today are posted at the TAMU eLearning site. - Monday, January 26
- We discussed the significance of Lebesgue's six axioms for integrals. We saw that Lebesgue's idea of reducing the problem of integration to the task of integrating functions taking the two values 0 and 1 leads to the problem of defining the measure of general sets. We observed that the definition of the measure of an open subset of the real line is forced upon us by the condition of countable additivity; taking complements then determines the measure of closed sets. We rephrased the definition of outer measure of a set E as the infimum of the measure of open sets containing E.
- Wednesday, January 21
- The first class meeting was an introduction to and motivation for the Lebesgue integral. We saw an example of a bounded function that is not Riemann integrable but is Lebesgue integrable (the Dirichlet function); we stated Lebesgue's theorem that a bounded function on an interval [a,b] is Riemann integrable if and only if the set of discontinuities of the function is a set of measure zero; we looked at some examples of sets of measure zero; and we defined Lebesgue outer measure.
- Tuesday, January 20
- The first-day handout is available, and the first assignment is posted.

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