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MAT337H1 (2)
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Lecture

Entire Course Notes Summary & Lectures

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Department
Mathematics
Course Code
MAT337H1
Professor
I.M Sigal

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INTRODUCTION TO REAL ANALYSIS (MAT337HS)
I. M. SIGAL, TAS BEN RIFKIND AND ANDREW STEWART
Anyfeedbackon lectures, tests and grading will begreatly appreciated!
Place and time: M4 – 5 pm, W2–4pm in UC A101
Course description: The goal of this course is to explain key concepts of Real Analysis with the view at applications.
The course is about the same level as MAT357, but while MAT357 deals mainly with theory,the presentcourse aims
at developing interesting applications.
Syllabus:
*Review: Sequences, series, functions (limits and continuity,differentiation and integration)
*Vector and normed spaces
*Convergence
*Spaces of functions
*Orthonormal sets and Fourier series
*Compactness
*Calculus of vector functions
*Contraction mapping principle
*Ordinary differential equations: Existence and uniqueness of solutions
*Optimization
Textbook:
[DD] Kenneth R. Davidson and Allan P.Donsig, Real Analysis and Applications, Springer, 2010.
Tests
Quizzes, midterm test, final test.
Quizzes and midterm and final exams will beon the material covered in the lectures. All the problems for the
quizzes and midterm exam most of the problems for the final test will bemodifications of the problems from the
homework. The problems in quizzes and exams will involveproofs.
Quizzes will last 25 min each.
Wewill havesix quizzes: Jan 17 (changed to 19), 31, Feb 7, and Feb 28, March14, 28 (changed to March7,
21, April 4).
Midterm: February 16.
Marking scheme
Breakup of the grade:
Quizzes/ midterm/ final test 35%/35%/30% .
Homework problems and supplementary material
The list of topics covered, homework problems and material whichis not found on the book (supplementary
material) will beposted on eachThursdayon the webpage
http://www.math.utoronto.ca/sigal/RealAnalNotes.pdf
Tutorials:
Before or after eachquiz, during the class time, the TAwill haveatutorial on the last week problems. (This will
takeplace on all quiz days.) On Mondays without quizzes, Iwill review problems on Wednesdays, preceding quizzes.
Please, raise problems you havedifficultywith.
Office hours:
TAwill hold office hours every week. On Fridays before aquiz TAwill hold twooffice hours from 11-1. On
other Fridays there will beone office hour from 11-12. If these times do not work for you, you can email TA
Date:April 1, 2011.
1
www.notesolution.com
2I. M. SIGAL, TAS BEN RIFKIND AND ANDREW STEWART
(b[email protected]) and he will try to set up another meeting time. The office hours will beheld in BA026
(Bahen).
Email help: Ican answer straightforward questions byemail, but Iwill not work through problems.
Lecture material
Review:
Section 1.2, Chapters 2, 3, 4and 5of [DD] (Sequences, series, Rn,limits and continuityof functions).
Chapter 6and Section 12.2 of [DD] (Differentiation and integration).
New material:
Sections 7.1,7.2, 7.4 of [DD]. (Vector spaces, norms, convergence (point-set topology), inner products).
Section 8.2 of [DD]. (Spaces of continuous functions, completeness).
Section 7.5 of [DD]. (Orthonormal sets).
Sections 7.5, 7.7 of [DD]. (Projections, Orthonormal expansions).
Sections 7.6 ,14.5 of [DD]. (Fourier series).
Part of Section 11.1 and Sections 12.1, 12.3 -12.5 of [DD] (Contraction principle and Differential equations).
Material: (Parts of)Sections 16.1, 16.3, 16.4 of [DD] (Convex sets).
(Parts of)Sections 16.5, 16.6, 16.9 of [DD] (Convex functions and optimization).
Homework for the course (from the textbook [DD])
Page 15: Ex A; Page 19, Problems A, D; Page 22, Problems A, B; page 31: Ex A, C.
Page 42, Problems A, B; Pages 51-52, Problems A, C-G, J, K; Page 55, Problems A, B, D.
Pages 71-72, Problems A, C, D, G, H, M; Page 76, Problem A; Pages 79-80, Problems B, D, E-G.
Page 98, Problems A, E, I, N; Page 102, Problems A, H, L; Page 108, Problems D, H; Page 112, Problems Ca, J, K;
Page 300, Problems A-D.
Page 117, Problems C, D, F, G, I; Page 119, Problems A-D, F-H, M, N; Page 127, Problems A-D, F, I-L; Page
132, Problems A, D; Pages 135-136, Problems A-D, E(b), F(a), G(a), I-L; Page 141, Problems B-D, F, G; Page
149: Ex A, B.
Page 251, Problems E, F, H, I; Page 297, Problems A, B, D; Page 304, Problem A, C; Pages 308-309, Problems A,
B, Da, E, Fa, H; Page 315, Problem H.
Pages 454, Problems A, B(a), F; Page 466, Problems A, B, C; Page 472-473, Problems A-D, F, I; Pages 476, Problems
A, B, H; Page 497, Problems A, B.
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INTRODUCTION TO REAL ANALYSIS 3
1. Review: Properties of limits and series
(a) If the sequence {an}
n=1 is convergent, then the set {an,nN}is bounded;
(b) lim αan=αlim an;
(c) lim(an+bn)lim an+lim bn;
(d) lim(anbn)lim anlim bn;
(e) lim(an/bn)lim an/lim bn;
(f)Monotonically increasing sequence whichis bounded above, converges;
(g) Every bounded sequence of real numbers has aconvergentsubsequence;
(g) Every Cauchysequence is bounded.
(a) If the series P
n=1 anconverges, then lim an=0;
(b) Absolutely convergentseries is convergent;
(c) If |an|bn,nand if P
n=1 bnconverges, then so does P
n=1 anand P
n=1 anP
n=1 bn;
(d) The following statements are equivalent
i) the series P
n=1 anconverges;
ii) limk→∞ P
n=kan=0;
iii) limk,l→∞ Pl
n=kan=0.
Theorem 1.1. The spaceRis complete.
Theorem 1.2. The spaceRnwith the standardnorm kxk:= (Pix2
i)1/2is complete.
2. Vector Spaces, Norms, Inner Products
In this lecture wereview briefly background material related to linear (or vector) spaces. Weintroduce the simplest
and most commonly used spaces, Banachand Hilbert spaces, and describetheir most importantexamples.
2.1. Vector spaces. Webegin with the key definitions. Avector space,V,is acollection of elements, denoted,
u, v, ...,for whichthe operations of addition, (u, v)u+vand multiplication bya(real or complex) number,
(α,u)αu,are defined in suchawaythat
u+v=v+u(commutativity)
u+(v+w)=(u+v)+w(associativity),
u+0=0+u=u(existence of zero),
α(βu)=(αβ)u,
(α+β)u=αu+βu
0v=0,
1v=v.
Wealso denote v:= (1)v.Elements of avector space are called vectors.As will beclear from the context most
of the vector spaces weconsider in these lectures are defined for multiplication byreal or complex numbers (they are
said to bevector spaces over real/complex numbers).
Here are some examples of vector spaces:
(a) Rn={x=(x1, ..., xn)|<xj<j}the Euclidean space of dimension n;
(b) C(Ω) the space of continuous real (or complex) functions on Ω, where is either either asubset of Rnor
Rn;
(b) Ck(Rn)the space of ktimes continuously differentiable real (or complex) functions on Rn,k=1, . . . .
The addition and multiplication byreal/complex numbers in these spaces is defined in the pointwise way:
(x+y)j=xj+yjand (αx)j=αxjj
and
(f+g)(x):= f(x)+g(x)and (αf)(x):= αf(x)x.
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Description
INTRODUCTION TO REAL ANALYSIS (MAT337HS) I. M. SIGAL, TAS BEN RIFKIND AND ANDREW STEWART Any feedback on lectures, tests and grading will be greatly appreciated! Place and time: M 4 5 pm, W 2 4 pm in UC A101 Course description: The goal of this course is to explain key concepts of Real Analysis with the view at applications. The course is about the same level as MAT357, but while MAT357 deals mainly with theory, the present course aims at developing interesting applications. Syllabus: * Review: Sequences, series, functions (limits and continuity, dierentiation and integration) * Vector and normed spaces * Convergence * Spaces of functions * Orthonormal sets and Fourier series * Compactness * Calculus of vector functions * Contraction mapping principle * Ordinary dierential equations: Existence and uniqueness of solutions * Optimization Textbook: [DD] Kenneth R. Davidson and Allan P. Donsig, Real Analysis and Applications, Springer, 2010. Tests Quizzes, midterm test, nal test. Quizzes and midterm and nal exams will be on the material covered in the lectures. All the problems for the quizzes and midterm exam most of the problems for the nal test will be modications of the problems from the homework. The problems in quizzes and exams will involve proofs. Quizzes will last 25 min each. We will have six quizzes: Jan 17 (changed to 19), 31, Feb 7, and Feb 28, March 14, 28 (changed to March 7, 21, April 4). Midterm: February 16. Marking scheme Breakup of the grade: Quizzes midterm nal test 35%35%30% . Homework problems and supplementary material The list of topics covered, homework problems and material which is not found on the book (supplementary material) will be posted on each Thursday on the webpage http:www.math.utoronto.casigalRealAnalNotes.pdf Tutorials: Before or after each quiz, during the class time, the TA will have a tutorial on the last week problems. (This will take place on all quiz days.) On Mondays without quizzes, I will review problems on Wednesdays, preceding quizzes. Please, raise problems you have diculty with. Oce hours: TA will hold oce hours every week. On Fridays before a quiz TA will hold two oce hours from 11-1. On other Fridays there will be one oce hour from 11-12. If these times do not work for you, you can email TA Date: April 1, 2011. 1 www.notesolution.com
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