# Math 446, Principles of Analysis I, Fall 2008

Harold P.
Boas

- Sunday, December 7
- I have posted solutions to the final exam.
- Friday, December 5
- The final exam was given.
- Tuesday, December 2 (the last class meeting)
- I have posted solutions to the final assignment at the TAMU eLearning site. The homework averages are posted there too; I averaged the best 9 of the 11 homeworks.

In class, we used a diagonal argument to prove one direction of the Arzelà–Ascoli theorem.

Reminder: the final examination is scheduled for 12:30–14:30 on Friday 5 December. - Tuesday, November 25
- We discussed the notion of equicontinuity and stated the Arzelà–Ascoli theorem.
- Thursday, November 20
- We deduced the polynomial version of the Weierstrass approximation theorem from the trigonometric version and obtained as a corollary that the metric space C[0,1] is separable. We sketched the proof of the trigonometric version of the Weierstrass approximation theorem and filled in some of the details.
- Tuesday, November 18
- We proved that the metric space C[0,1] of continuous real-valued functions is connected, complete, and not compact.
- Thursday, November 13
- We proved the lemma from last time that if a real-valued function f on the interval [0,1] is the pointwise limit of continuous functions, then the inverse image under f of every open interval is an F
_{σ}set. We also checked the equivalence of three versions of the Baire category theorem.

The assignment for next Thursday is available. This assignment is the last one for the semester. - Tuesday, November 11
- We proved, modulo a key lemma, that if E is the set of discontinuities of a function on [0,1] that is a pointwise limit of continuous functions, then the set E is a countable union of closed, nowhere dense sets (thus a set of first Baire category). In particular, the set of points where such a function is continuous is a dense set.
- Thursday, November 6
- We proved the Baire category theorem and had an initial glimpse of the application that "most" continuous functions are nowhere differentiable.

The assignment for next Thursday is available. - Tuesday, November 4
- The graded exams were returned; the median score was 92.

Class started late because of the fire alarm. We discussed the notion of nowhere dense sets and stated the Baire category theorem. - Thursday, October 30
- The second exam was given, and solutions are available.

The assignment for next Thursday is available. - Tuesday, October 28
- We continued discussing uniform continuity. In particular, we proved that a continuous function on a compact metric space is automatically uniformly continuous. Solutions to the two most recent homework assignments now are available at the TAMU eLearning site.
- Thursday, October 23
- We continued the discussion of compactness in metric spaces, adding the equivalent property that a nested sequence of closed sets has nonempty intersection with a compact set S if each individual closed set has nonempty intersection with S. We also proved that a continuous real-valued function on a compact metric space attains a maximum value and a minimum value, and we recalled the notion of uniform continuity.

The assignment for next week is to study for the examination to be held on Thursday, October 30 over Chapters 4–7 of the textbook. - Tuesday, October 21
- We discussed three equivalent notions of compactness in metric spaces: completeness together with total boundedness; sequential compactness; and the property that every open cover admits a finite subcover.

Reminder: the second exam is scheduled for Thursday, October 30. - Thursday, October 16
- We discussed the notion of completeness in general metric spaces. We saw that completeness is equivalent to a nested set property and also to a suitably rephrased version of the Bolzano-Weierstrass theorem.

The assignment for next Thursday is available. - Tuesday, October 14
- We discussed the concept of a totally bounded metric space and the generalization of the Bolzano-Weierstrass theorem saying that a metric space is totally bounded if and only if every sequence in the space has a Cauchy subsequence.
- Thursday, October 9
- We discussed the concept of connectedness for metric spaces.

The assignment for next Thursday is available. - Tuesday, October 7
- We discussed the notions of isometry and homeomorphism for metric spaces, and we looked at some properties of the space C[0,1] of continuous functions. In particular, we saw that the metric structure on C[0,1] is compatible with the vector-space structure. We also saw that the uniform metric on C[0,1] and the L
^{2}metric on C[0,1] produce metric spaces that are not homeomorphic to each other (since the two spaces have different sets of Cauchy sequences). - Thursday, October 2
- We discussed how the notions of open sets and closed sets in metric spaces interact with the notion of convergence of sequences, we looked at three equivalent ways to phrase the notion of continuity of a function between metric spaces, and we applied these ideas to study some examples of functions between metric spaces. We also used the completeness axiom to prove that the real numbers cannot be decomposed as the union of two disjoint, non-empty, open subsets.

The assignment for next Thursday is available. - Tuesday, September 30
- We discussed, in the setting of metric spaces, topological notions such as neighborhoods, open sets, closed sets, interior, closure, and density.

The assignment due next class is exercises 8 and 11 on page 55 and exercise 19 on page 57. - Sunday, September 28
- Solutions to the first exam are available.
- Thursday, September 25
- The first exam was given.
- Tuesday, September 23
- We reviewed for the exam to be given next time.
- Thursday, September 18
- We finished the discussion of the inequalities of Hölder and Minkowski.

The assignment is to prepare for the examination to be given next Thursday (September 25) on Chapters 1–3 of the textbook. - Tuesday, September 16
- There is now a
**help session**for Math 446 on Monday and Wednesday mornings from 9:30 to 12:00 in Blocker 605.

In class, we discussed the Cauchy-Schwarz inequality, Hölder's inequality, and Young's inequality. - Thursday, September 11
- I posted the assignment due next Thursday.

In class, we discussed the notion of limits in metric spaces, and we looked at the definition and examples of norms.

The grader has prepared solutions to the assignment that was due on September 4, and registered students can access the solutions through the TAMU eLearning site. - Tuesday, September 9
- We proved the continuity of the Cantor function using the ε–δ definition of continuity. (There is a different proof in the textbook.) Then we began a discussion of metric spaces by considering the definition and some examples.
- Thursday, September 4
- I posted the assignment due next Thursday.

In class, we discussed the construction of the Cantor set (both from the geometric point of view and from the point of view of ternary expansions), the uncountability of the Cantor set, and the Cantor function. - Tuesday, September 2
- We discussed the notion of cardinality of infinite sets, the equivalence of the natural numbers and the Cartesian product of two copies of the natural numbers, the non-equivalence of the natural numbers and the real numbers, the non-equivalence of any set with its power set, the Bernstein equivalence theorem, and (briefly) the continuum hypothesis.

As previously announced, the assignment due next class is posted online. - Friday, August 29
- Yesterday in class we discussed three equivalent ways to define limsup and two equivalent ways to define continuity. We proved in detail that the so-called ruler function is discontinuous at every rational number yet continuous at every irrational number. The assignment due next Thursday is posted online.
- Tuesday, August 26
- We discussed the definition and the properties of the real numbers. In particular, we talked about the following completeness properties of the real numbers:
- Every Cauchy sequence of real numbers converges (to a real number).
- Every non-empty set of real numbers that is bounded above admits a
*least*upper bound (called the supremum). - Every bounded monotonic sequence of real numbers converges (to a real number).
- Every nested sequence of closed, bounded intervals has a non-empty intersection.
- Bolzano-Weierstrass theorem: every bounded sequence of real numbers has a convergent subsequence.

- Section 500 should do exercises 1, 3, and 4 on page 4 of the textbook to hand in on Thursday;
- Section 200 should hand in next time a discussion of the following question: are the five completeness properties equivalent to each other?

- Monday, August 25
- This site went live today. The first-day handout is available online and also in a printable format.

These pages are copyright © 2008 by Harold P. Boas. All rights
reserved.