Topics in One Variable Analysis (Review)

Syllabus (PDF)

COURSE LOG

F 8/25 Norms in R^n and function spaces

Tu 8/29 Norms, completeness, equivalence of norms--examples

Cauchy-Schwarz inequality for integrals

Lecture 1-Norms, completeness, Bolzano-Weierstrass

First Homework Set

(includes 3 HW problems, due Tu 9/5)

Th 8/31 Pointwise, uniform and L1 convergence--implications, examples

Bolzano-Weierstrass theorem (in R and R^n)

Tu 9/5 open sets, closed sets, closure, continuity, homeomorphism (in R^n)

Lecture 2-continuity, uniform continuity, basic topology

Th 9/7 accumulation points, distance from point to set

Second Homework Set (due 9/14)

HW2 solutions

How to invert a linear operator

Tu 9/12 uniform continuity--extension to the closure; Lipschitz and Hoelder conditions

Th 9/14 dense subsets, operator norm/ compactness in R^n: homeomorphisms, uniform continuity, proper maps

Third problem set

HW3 (due 9/21): problem 1(i)(ii)(iii), problem 2(i)(ii), Problem 3(i)(ii), Problem 5(iii)

Tu 9/19 Compactness (cont'd): intersection of decreasing sequence of compact sets, Heine-Borel theorem.

Lecture 3: compactness, extension, convex functions

Th 9/21 Lebesgue number of a covering/Weierstrass theorem on series of cont. functions/Extension of continuous functions (Tietze)

HW 4 (due 9/28)--from Problem set 3: 1(v), 2(iv)(v)(vi), 6(iii), (v), 7(i) (ii)

Tu 9/26 Convex functions (continuity), monotone functions (countable discontinuity set)/Ascoli-Arzela theorem (start)

Arzela-Ascoli notes

(when reading this, think only of the case when the domain is a subset of R^n, and the range some R^p.)

Th 9/28 Arzela-Ascoli (cont.)

Tu 10/3 Problem session

Th 10/5 FALL BREAK (no lecture)

Tu 10/10--Midterm

midterm-solutions

Th 10/12 Differentiable functions and maps/ directional derivatives/ examples, scalar mean value theorem

Differentiable maps

HW 5 (due 10/19) From Fleming: p.79: 3/ p.88: 4abc (compute the differential, not the gradient), 5, 7(b), 8, 10

Tu 10/17, Th 10/19 Mean value theorems/ partial derivatives and differentiability/ differential of composition

HW 6 (due 10/26) Exercise 2 and exercise 3 from the handout "Differentiable maps"

From Fleming: 3.4 (p. 98): 4, 5, 7(c)

Tu 10/24, Th 10/26: Schwarz's theorem, functions and maps of class C^k, Taylor's theorem

Taylor's formula with remainder

HW 7 (due 11/2) p. 98 no. 2/ p. 106 no. 4,5,7

Tu 10/31, Th 11/2: Non-degenerate critical points, Hessian criteria for local max/min, convex functions

Practice problems (p.118) 6, 7, 11, 12.

The Hessian and Convex Functions

Tu 11/7: Implicit function theorem (proof)/ discussion of problems

Solutions to selected homework problems

Th 11/9: Second midterm (included: material from 10/12 to 10/31)

Midterm 2 (with solutions)

11/14, 11/16: regular values, hypersurfaces, tangent space, Lagrange multiplier, InvMT (start))

11/21: InvMT: Banach's fixed point theorem, Lipschitz perturbations of invertible linear maps, property of C1 maps/ HW 8 due

Homework set 8 (6 problems)

HW8 solutions

Inverse Mapping Theorem

11/24: Thanksgiving (no classes)

11/28- Inv MT: differentiability of the inverse, inverse is C^k, implicit map theorem (submersions)- HW 9 posted

HW 9: Exercises 1, 3 and 4 from the handout "Inverse Mapping Theorem", and the three problems from Fleming on p.14/15

of the same handout.

(Read the handout up to Proposition 4 on p. 12)

11/30: Immersions and surfaces in R^n/ sequences and series of differentiable maps

Spaces of differentiable maps

12/5 (last day): sequences and spaces of differentiable maps; HW 9 due

HW9 solutions

12/12: Final Exam (comprehensive): 10:15--12:15.

For next semester: local existence/uniqueness for vector fields, mountain pass lemma