MATH 447- Advanced Calculus I- Fall 2016- A. FREIRE

(or: ANALYSIS IN R^n)

Syllabus

Topics

W 8/17  Normed vector spaces and Banach spaces: examples. R^n with different norms, bounded real-valued functions from a set,
continuous functions from the unit interval (supremum norm.)

First Problem Set
solutions-HW1

(HW 1- F 8/26- problems 1-4)

F 8/19 Banach spaces: further examples/ closed subspaces/ L^p spaces: Minkowski inequality (start)

M 8/22 Minkowski and Hölder  inequalities: proof.

W 8/24 Open sets, closed sets, continuity/ topological spaces, metric spaces (definitions)

F 8/26 Comparison of uniform, pointwise and L1 convergence/ continuity does not reduce to lines

M 8/29 Continuity of composition/uniform continuity/Lipschitz and Hoelder conditions.

W 8/31 accumulation points, closure/extension of unif. continuous maps/

F 9/2 comments on HW1/ homeomorphisms (start)

Second Problem Set
HW 2 (due Friday 9/2): Problems 5, 6 from the first Problem Set and Problems 1, 4 and 6 from the second Problem Set.
solutions-HW2

M 9/5 LABOR DAY (no classes)

W 9/7 homeomorphisms-examples/BolzanoWeierstrass thm/ compact sets are closed (in Hausdorff spaces), bounded (in metric spaces)
Practice problems: 2.8--2, 4, 5/ 2.9--3, 4, 5

F 9/9 Heine-Borel theorem/ applications of compactness (continuous image of compact, uniform contimuity, continuous bijections are homeomorphisms)

HW 3 (due F 9/16): Problem set 2: 2, 3, 7, 8, 10.
Problem Set 2B (with hints)
solutions-HW3

M 9/12 Compactness, metric spaces--examples (equivalence of norms, problem 9, practice problems)

W 9/14 connectedness and homeomorphism (section 2.7)
Notes on connectedness (includes 8 problems)
HW 4, due 9/23: problems 1, 2, 3, 4, 8 in these notes.
HW4 solutions

F 9/16 connectedness and homeomorphism (cont.)

M 9/19 closed hyperplanes and bounded linear functionals (handout in `topics' link.)

W 9/21 locally compact Banach spaces are finite dimensional (see handout in link "topics", above).

F 9/23 spaces of continuous functions: Stone-Weierstrass theorem (handout in `topics' link).

M 9/26 Stone-Weierstrass theorem, cont'd/ Bernstein polynomials
HW 5, due 9/30: [Fleming], p.66, no. 3/ no. 4(c), (d)
Handout "locally compact Banach spaces: exercise 1, exercise 4
Handout "Stone-Weierstrass theorem": exercise 1
HW5solutions

W 9/28 Ascoli-Arzela  theorem (handout in `topics' link.)

F 9/30 Arzela-Ascoli: proofs.

Read: linear functions (3.2) in [Fleming]

M 10/3 Arzela-Ascoli remarks/ directional derivatives (examples)
HW 6, due Monday 10/10: Exercises 1-7 in the Arzela-Ascoli notes.

W 10/5: differentiable functions/ continuity, mean value theorems
Differentiable Maps (in progress): exercises 1 to 6 are HW 7, due Wednesday 10/19.

F 10/7: Fall Break (no lecture)

M  10/10: C^1 functions

W  10/12 Chain Rule

F  10/14 Differential of inversion/ From infinitesimal to local

M 10/17 Strong differentiability

W 10/19 Problem session
HW7-solutions

F 10/21 Schwarz's theorem

M 10/24:
Included: all the handouts posted, including HW solutions (be sure to check you have the latest posted version)
Sections in Fleming's text: 2.4 to 2.10, 3.1, 3.3, 4.3, 4.4.
You may be asked for definitions and statements of theorems (in addition to short proofs.)
midterm solutions

W 10/26 Sets of discontinuity and Baire's theorem
Baire Category Notes (5 problems) (the problems are HW8, due Friday 11/4)
HW 8 solutions

F 10/28 Baire Category arguments

M 10/31 Higher differentiability, Taylor approximation

W 11/2 Local max-min and the Hessian/ convexity (start)
The Hessian and Convex Functions

F 11/4 Convexity and differentiability
(including section 3.6 in [Fleming])
HW9 (due 11/11): from [Fleming]: 4, 7 (p.99); 5, 7 (p.106); 7, 9 (p.118)
HW 9 solutions

M 11/7 Inverse Function Theorem (start)
Inverse Function Theorem

W 11/9 Inverse Function Theorem (end)

F 11/11 Implicit Function Theorem

M 11/14 Some applications of the IFT
HW 10: the two exercises in the Inverse Function Theorem handout (see link for 11/7) plus
three problems in [Fleming]: 7 and 8 on p. 147, 11 on p.160
HW 10 solutions

W 11/16 Surfaces in R^n; tangent spaces

F 11/18 Convergence of differentiable maps
Spaces of Differentiable Maps (inc. 4 problems=HW 11)
(includes solutions to HW 11)

M 11/21 HW10 due/ Spaces of differentiable maps/ compact linear maps

W 11/23 Lebesgue's theorem (monotone functions are differentiable a.e.)
Lebesgue's theorem on monotone functions

F  11/25  Thanksgiving

M 11/28  Lebesgue's theorem (HW 11 due)

Final Exam--Monday, 12/5, 8-10AM (comprehensive)
Final Exam (with solutions)

Material included: all the online handouts (except the 11/23 one), homework problems,
sections in Fleming's text:
2.8 to 2.11
3.1 to 3.6
4.3 to 4.7
(closed book, closed notes.)