Math 347 – Advanced Real Analysis
Review for Test 1
Test 1 will consist in 5 problems.
Review the theory for each section, including the proofs for the theorems listed, and homework problems.
Ch. 3 and Ch. 5: Sequences and Basic Topology
- Def. distance; metric space; bounded sets; Euclidean metric;
- Def. open/closed sets and balls; complementary notions;
- Properties (“topology”): Open sets closed under union and finite intersection (w. proofs);
- Closed sets are closed under finite union and (arbitrary) intersections
- A bounded & closed set => max/min exists (l.u.b. belongs to the set);
- Convergence of sequences to a 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) and lim:
- Operations on convergent sequences: +,-,x,: & lim “commute”;
- Convergence compatibility w. order: anbn=>lim anlim bn;
- Increasing and decreasing functions;
- Monotonic Sequence Theorem: bounded & monotonic + complete => convergent;
- Cauchy sequences: def. & properties w. proofs. (Convergent => Cauchy=> bounded);
- Completeness: def., properties (complete => a closed subspace is complete); R is complete;
- Compactness: def. with sequences or with open covers; properties:
- Closed subset in compact is compact
- Compact => bounded (pf.) & nonempty intersection property & closed (pf.);
- Accumulation points: def.; example: infinite subsets of a compact space;
- Closure of a set: def. ; cl(S)=accumulation points S.
- Fact: in En closed & bounded => compact;
- Connectedness: def.; property: a connected union of connected sets is connected (pf).
- Example: R (P connected subset interval).
Ch.6 Continuous Functions
- Limit of a function at a cluster point;
- Continuity: local & global, using epsilon-delta, sequences or open/closed sets equivalent formulations;
- Operations preserving continuity: +,-,x, and composition of fn. (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.);
- Classifying discontinuities: 1st type, 2nd type; discontinuities of monotone functions
- Uniform continuity: def.; example;
- Uniform continuity => continuous (pf.); counterexample for the converse xn;
- Continuity + compact domain => uniform continuous;
- Lipshitz function => uniform continuous
Math 347 – Advanced Real Analysis
Review for Test 2
Test 1 will consist of 5 problems. Review the theory for each section, including the proofs for the theorems listed, and homework problems.
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; chain rule
- Fermat’s Th, Role’s Th. (proofs)
- Mean Value Theorem for Derivatives
- Zero derivative => constant function (pf.);
- Monotone functions: f diff. => f increasing <=> f’ nonnegative;
- Higher derivatives, higher order approximations;
- Indeterminate forms and L’Hospital’s rule
- Inverse Function Theorem (f’(p)0 => f-1 differentiable etc.)
Ch. 13 Functions of several variables
- Linear algebra review (vector spaces and linear transformations)
- Normed vector spaces (=> metric spaces)
- Convergence and continuity
- Differentiable functions and the derivative as a linear transformation (functional)
- Partial derivatives as components of the derivative
- Properties of the derivative (rules for derivatives, including the chain rule)
- Taylor expansions
Ch. 8 The Integral
- The Riemann Integral: partitions, Riemann sums, R. Integral
- Cauchy criterion for integrability
- Properties:
1) Integrable => bounded, 2) continuous => integrable,
3) Bounded & finitely many discontinuities => integrable
- Properties of the Riemann integral:
1) Linear functional, increasing (positive functional), bounded (| f||f|).
2) Additive with respect to the domain,
3) product Integrable functions is integrable; continuous o integrable => integrable
- Fundamental Theorem of Calculus
- Rieman-Stieljes integral (upper/lower RS relative to a measure (x)dx; RS-integrable)
- Relation with RI: (x)=x => RSI=RI; “change of variables” (f d=f ’ dx)
- Integration by parts formula
- Bounded variation
- Continuous + Bounded Variation => RS-integrable (fixed domain)
Math 347 – Review for Test 3
Test 3 will consist of 5 problems (possibly including some theoretical questions: definitions and properties)
Review the theory for each section, including the proofs for the theorems listed.
Review the homework problems.
Ch. 9 Sequences and series of functions
- Sequences of functions, pointwise and uniform convergence, Cauchy sequences
- Continuity and uniform convergence
- Integrability and uniform convergence
- Differentiability and uniform convergence
- Interpretation of the above properties as commutative properties (interchanging limits and integration, differentiation)
- Series of functions, sum of the series, pointwise and uniform convergence
- The Weierstrass M-Test
- The Weierstrass approximation theorem
- Applications to power series: radius and interval of convergence (uniform on compact intervals), term-by-term integration and differentiation (from Ch. 10 Power Series)
Ch. 11 Applications of analysis to differential equations
- Differential equation, initial value problem, Lipschitz condition
- Picard’s existence and uniqueness theorem, Picard’s iteration method (Fundamental Th. of Calculus & iteration starting with the initial condition approximation)
- The methods of characteristics (examples & homework problems)
- Power series method (ordinary points)
Ch. 12 Introduction to Harmonic Analysis
- Complex exponential functions (characters): orthonormal basis in C(S1,C) relative the “dot product” <f,g>
- Fourier coefficients = components of f in the Fourier basis of exponentials
- Computing the Fourier coefficients (example)
- Fourier transform: definition and properties (linear, intertwines differentiation and multiplication by the independent variable)
Math 347 – Review for Final Test
(In addition to the reviews for Tests 1, 2, 3)
Final Test will consist of 10 problems. One page of notes is allowed.
Review the theory for each section, including the proofs for the theorems listed.
Review the homework problems.
12.4 Fourier methods in the theory of differential equations
- Method of separation of variables
- The Dirichlet problem on the disc: general solution and finding the Fourier coefficients from an initial condition
- The wave equation: general solution and finding the coefficients from the initial conditions
Ch. 15 A Glimpse of Wavelet Theory
- Localization in the time and space variables
- A custom Fourier analysis: characteristic function, scaling function, wavelet
- Space translations, dilates; the vector spaces spanned by the translates and dilates of a wavelet
- The Haar wavelet basis
- Axioms for a multi-resolution analysis
(Useful practice for preparing the theory)
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?