# Math 347 Advanced Real Analysis

Math 347 – Advanced Real Analysis

Review for Test 1

Test 1 will consist in 5 problems, 2 sets of theoretical questions, and an extra credit problem.

Review the theory for each section, including the proofs for the theorems listed, and homework problems.

Ch.1 Notions from Set Theory

- Sets and elements; notation; quantifiers and negation;

- Operations with sets: union, intersection, difference, product;

- De Morgan laws (w. proof);

- Family of sets;

- Functions; 1:1 and onto; direct image and inverse image;

- Identity function, inverse function; characterization (1:1 and onto);

- Finite sets: def., 1:1=> onto; infinite set: def., it is equivalent to a proper subset.

Ch.2 The Real Number System

- Def.: complete ordered field;

- Field properties: def. and consequences (w. elementary proofs: inverses, opposites and distributivity);

- Examples: Q, R, C (ex.8);

- Def. order; Order Axiom: R+ & the associated order; consequences:

- Trichotomy, transitivity, compatibility with + and x.

- Absolute value: def. and properties;

- Bounds and least upper bounds;

- PVII: Least Upper Bound Property/axiom: def. & consequences

- Using opposites => existence of g.l.b. (pf.)

- LUB1 (N is unbounded) w. proof;

- It is a completeness property/axiom.

- Decimal numbers: def., correspondence: finite-> Q & infinite -> l.u.b;

- Existence of square roots;

- Understanding that the Real Number System is unique modulo isomorphism:

- Algebraic axioms + square roots => R+ uniquely determined by (R,+,x);

- Decimal reps => concrete realization.

Ch.3 Metric Spaces

- Def. distance; metric space; bounded sets;

- Euclidean metric, Schwartz inequality & interpretation;

- Def. open/closed sets and balls; complementary notions;

- Properties (“topology”):

- Open sets closed under union and finite intersection (w. proofs);

- Separation properties;

- P: bounded & closed => max/min exists (l.u.b. belongs to the set);

- Convergence to limit:

- Def. eps & open sets; properties w. pfs.

- Limit is unique; conv. is hereditary; f. many terms don’t count;

- Convergent => closed and bounded;

- Properties of (R,+,x): topological field:

- Operations on convergent sequences: +,-,x,: & lim “commute”;

- Convergence compatibility w. order: anbn=>lim anlim bn;

- Increasing and decreasing functions;

- P (Monotonic Sequence Th.): bounded & monotonic + complete => convergent;

- Cauchy sequences: def. & properties w. pfs. (Convergent => Cauchy=> bounded);

- Completeness: def., properties (complete => a closed subspace is complete);

- Fact: R is (sequentially) complete;

- Compactness: def., properties:

- Closed subset in compact is compact (pf.)

- Compact => bounded (pf.) & nonempty intersection property (pf.) & closed (pf.) & complete (pf.);

- Cluster points: def.; example: infinite subsets of a compact space;

- Closure of a set: def. ; cl(S)=cluster points  S.

- Fact: in En closed & bounded => compact;

- Connectedness: def.; property: a connected union of connected sets is connected (pf).

- Example: R (P: a connected subset is an interval).

Math 347 – Advanced Real Analysis

Review for Test 2

Test 1 will consist in 5 problems, 2 sets of theoretical questions, and an extra credit problem.

Review the theory for each section, including the proofs for the theorems listed, and homework problems.

Ch.4 Continuous Functions

- Continuity: local & global, using epsilon-delta, sequences or open/closed sets equivalent formulations;

- Examples: identity fn. and constant functions are continuous (pf.);

- Operations preserving continuity: restriction, +,-,x, and composition of fn. (pf.)

- The extension problem: cluster points, limit of a function at a cluster point;

- Continuity rephrased using “the limit of a function” concept;

- Sequential continuity of a function: definition & equivalence to continuity of a function;

- Projections are continuous fns. (En->E);

- Vector valued functions: def., characterization of continuity (pf.);

- Continuous fn. preserve connectedness (pf.); Intermediary Value Th. (pf.)

- Continuous functions preserve compactness (pf.); consequences:

- Continuous fn. are bounded on compact sets (pf.);

- Existence of max & min of continuous fn. on compact sets (pf.);

- Uniform continuity: def.; example;

- Uniform continuity => continuous (pf.); counterexample for the converse;

- Continuity + compact domain => uniform continuous;

Ch. 5 Differentiation

- Definitions: derivative (local/global), differentiable fn., linearization;

- Differentiable => continuous (pf.), counterexample for the converse;

- General rules for derivative: d/dx & +,-, . commute (. is scalar multiplication by a constant);

- Product rule, quotient rule, power rule (proofs);

- Chain rule, (pf. for alternate credit (AC));

- Fermat’s Th, Role’s Th. (proofs)

- Mean Value Theorem for Derivatives (pf. AC)

- Zero derivative => constant function (pf.);

- Monotone functions: f diff. => f increasing <=> f’ nonnegative (pf.);

- Higher derivatives, higher order approximations;

- Taylor polynomial, remainder and series; examples & computing;

- Taylor’s Theorem (state and prove for AC);

Math 347 – Review for Final Test

(In addition to the reviews for Test 1 and Test 2)

Final Test will consist of 10 problems, including 4-5 questions. One page of notes is allowed.

Review the theory for each section, including the proofs for the theorems listed.

Review the homework problems.

Ch. 6 Riemann Integration

- Riemann sums and integral: def., example: characteristic functions are Riemann integrable (RI) (pf.);

- Properties of Riemann Integral: linearity, positivity, monotonic;

- Average value & RI (between min & max) (pf.);

- Cauchy criterion for RI (“Cauchy => convergent”);

- Step functions: def., integrability, density in the space of RI functions (pf.) (R. sum is  of a step fn.)

- RI => bounded; continuous fn. => RI;

- Additivity of  w.r.t. integration domain;

- Orientation of the domain: def. of oriented ;

- Pf.: |f|<M => I(f)<M(b-a)

- Fundamental Th. of Calculus

- Existence of antiderivatives (statement & proof AC)

- Evaluation Th. (statement & proof)

- Change of variable th;

- Logarithmic and exp. function: def., properties (w. pf.)

Ch. 7 Interchange of Limit Operations

- Integration & limit: the problem, poitwise convergence is not enough (example);

- Uniform convergence + continuity => integration & limit commute;

- Differentiation & limit: the problem, poitwise convergence is not enough (example – see continuity!)

- Uniform convergence of derivatives + convergence at a point => differentiation & limit commute;

- Infinite series: def.; properties: translating properties of convergence of sequences;

- Operations with series: translating limit laws for sequences (pf. for +,-)

- Absolute convergence: def.; abs. conv.=> convergence (pf.);

- Tests of convergence: Comparison Test , Ratio Test (statement & using); alternating series test (pf.);

- Series of functions:

- Series of fn. uniform convergent => Cauchy; cor.: |fn(x)|<an convergent => fn conv. abs. & unif.

- Integration and differentiation term-by-term theorem (statement);

- Power series: def., example: geom. series; radius and interval of convergence;

- Integrating & differentiating power series:

- Radius of convergence does not change (pf. if limit exists);

- Uniform convergence => int. & dif. term-by-term;

- Power series representations:

- Def., uniqueness (pf.), Taylor poly., series & remainder; f(x)=s(x) iff Rn(x,a)->0;

- Using Taylor’s Th. to prove f(x)=s(s) (Taylor series is the power series representation)

- Trigonometric functions:

- Def. & solving the DE using power series solutions;

- Properties: values at 0, derivatives, additive formulas (pf.)

Questions Ch.1 & 2

1) What is a function? What is the direct image? What is the inverse image of a set?

2) What is a family of sets?

3) What is the complement of a union?

4) What is a field?

5) What is a linear order?

6) What are the properties satisfied by R+, the positive elements of an ordered field?

7) What is the least upper bound of a set of real numbers?

8) Why do square roots exist?

9) What is a real number system?

Questions Ch. 3 (I)

1) What is a distance? What is a metric space?

2) What are open sets? What are closed sets?

3) Is a finite set in a metric space, a closed set?

4) What is a bounded set in a metric space?

5) When is a sequence convergent to a limit?

6) Do the field operations preserve convergence? In what sense?

7) Is convergence compatible with the order? In what sense?

8) Are monotonic sequences convergent? What else is it required to ensure convergence?

9) What are Cauchy sequences? Are convergent sequence Cauchy?

10) What is a complete space? Is a closed set complete?

11) What is a compact space? What properties does a compact set have?

12) Is a closed and bounded set compact?

13) What is “the nonempty intersection property”? What space/set satisfies this property?

14) What is a cluster point? 15) What is the closure of a set? What is the relation between the set of cluster points and the closure of a set?

16) What is a connected set? What are the connected sets of the real number system?

17) What’s a “connected intersection” of connected sets? What can be said about this intersection?

Questions Ch.4

1) What is a continuous function?

2) What are the operations preserving continuous functions?

3) What is the limit of a function at a point?

4) What is the “Extension problem”?

5) Is a function continuous at a point in the domain, which is not a cluster point?

6) Does the extension problem have a solution at a non cluster point? How many solutions are there at a cluster point?

7) When is a vector valued function continuous? Why?

8) What topological properties are preserved by the inverse image of a continuous function?

9) What are the properties preserved by the direct image of a continuous function? (Open, closed, convergent, complete, compact, connected)

10) What is the content of the Intermediate Value Theorem?

11) What is point-wise convergence?

12) Does point-wise convergence preserve continuous functions?

13) What is uniform convergence? What is a uniformly Cauchy sequence?

14) Does uniform convergence preserve continuous functions?

15) How is the “sup” distance (norm) defined, and for what type of functions?

16) When is C(E,E’), with the above distance function, a complete metric space?

Questions Ch. 5

1) What is a differentiable function?

2) Is a differentiable function continuous?

3) What is the derivative, as an operation on differentiable functions?

4) Are differentiable functions stable under +, -, x, / and composition?

5) What is the content of Fermat’s Th?

6) What is the content of Role’s Th.?

7) What is the Mean Value Theorem?

8) How does the derivative relate to increasing and decreasing differentiable functions?

9) What is the Taylor polynomial and remainder of a smooth function?

10) What is the content of Taylor’s Theorem and Remainder estimate?

Questions Ch. 6

1) How is Riemann integral (RI) defined?

2) What are the main properties of RI?

3) What is the content of the Average Value Th.?

4) What are step functions? In what sense they are “dense” within Riemann integrable functions?

5) What is the “oriented” Riemann integral?

6) What is a rough estimate for the RI of a bounded integrable function on a compact interval?

7) What is the content of the Fundamental Theorem of Calculus?

8) What is the fact behind the Change of Variables Formula, and what does it say?

9) How is the natural logarithm defined? How are the exponential and the natural base e defined?

## Questions Ch. 7

1) Do point-wise convergence and integration commute? (Is integration continuous w.r.t. point-wise convergence?)

2) Does uniform convergence of continuous functions preserve integrability?

3) When do limit and differentiation commute? (What kind of limit?)

4) When do limit and integration commute?

5) What are series? What is the sum of a series?

6) What are the operations preserving convergent series?

7) What is absolute convergence?

8) What is the Comparison Test? Ratio Test? Alternating Series Test?

9) What series of functions can be differentiated term-by-term, preserving convergence? (When and where is differentiation continuous? i.e. w.r.t. what convergence and at what series?)

10) What is a power series? What is the interval of convergence and radius of convergence of a power series?

11) In what sense is the sequence of partial sums of a power series convergent to its sum?

12) What is a power series representation of a function? How many are there?

13) What is the Taylor series of a function centered at a point? How is the remainder defined?

14) What does the Taylor Theorem say about power series representations? (Convergence and Remainder estimate)

15) How are power series used to solve differential equations (DE)?

16) How are differential equations used to define new functions?

17) What is the relation between the coefficients of a power series solution of a DE and the initial conditions of an initial value problem?

18) What topic from this course you (dis)liked the most?