Department

Mathematics

Course Code

MAT337H1

Professor

I.M Sigal

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.

www.notesolution.com

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,n∈N}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 an≤P∞

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 Ωis 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=αxj∀j

and

(f+g)(x):= f(x)+g(x)and (αf)(x):= αf(x)∀x∈Ω.

www.notesolution.com

Over 90% improved by at least one letter grade.

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California

Join OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.