PMath/AMath 331, Fall 2014 - Solutions for assignment 11
Handed out on Friday November 21; due on Friday November 28.
Topics: Pointwise convergence of functions, uniform convergence of functions.
Prob
Friday, November 6 Lecture 23 : More on continuous functions f : V1 into V2.
Expectations:
1. State the topological definition of continuity on S V .
2. Use the basic properties of continuous function
PMath/AMath 331, Fall 2014 - Solutions for assignment 10
Handed out on Friday November 14; due on Friday November 21.
Topics: Finite dimensional vector spaces, pointwise convergence of functions, unif
Wednesday, September 23 Lecture 5 : Subsequences and the Bolzano-Weierstrass
theorem.
Expectations:
1.
2.
3.
4.
5.
Define a subsequence of a sequence.
State the Nested interval lemma.
State the Bolzan
Friday, September 18 Lecture 3 : Basic properties of Limits.
Define bounded above, bounded below and bounded sets in .
Show that convergent sequences in are always bounded in .
Recognize that the limi
Wednesday, September 16 Lecture 2 : Limits
Objectives:
1. Define the limit L of a sequence cfw_xn in .
2. Use the definition of the limit to determine whether a sequence converges or not
in .
3. Use t
Monday, September 14 Lecture 1 : The Real numbers.
Objectives:
1.
2.
3.
4.
5.
Define the real numbers.
Define the real numbers are dense
Define countable sets and uncountable sets.
Provide examples of
Friday, September 25 Lecture 6 : Cauchy sequences.
Define a Cauchy sequence in .
Prove that every convergent sequence in is Cauchy.
Define a complete subset of .
State the Completeness theorem in (The
Monday, September 21 Lecture 4 : Supremum and infimum of a subset of .
Monotone sequences.
Define the least upper bound (supremum) of a set in .
Define the greatest lower bound (infimum) of a set in .
Wednesday, September 30 Lecture 8 : Applications: Contractive sequences : Error
estimation.
Expectations:
1. Estimate the error at the nth term of a contractive sequence of real numbers.
Once we have
Monday, September 28 Lecture 7 : Applications: Contractive sequences.
Expectations:
1. Define a contractive sequence.
2. Show that contractive sequences are Cauchy and so must converge in .
7.1 Defini
Friday, October 2 Lecture 9 : n - Review of the inner product spaces : The
Euclidean norm on n.
Define the Euclidean inner product on n.
Define an abstract inner product.
Define the Euclidean norm on
PMath/AMath 331, Fall 2014 - Assignment 9
Handed out on Friday November 7; due on Friday November 14.
Topics: Intermediate Value theorem, continuity in normed vector spaces, uniformly
continuous funct
PMath/AMath 331, Fall 2014 - Assignment 8
Handed out on Friday October 31; due on Friday November 7.
Topics: Lipschitz constant of matrix transformations, Contractions, properties of
continuous functi
PMath/AMath 331, Fall 2014 - Assignment 6
Handed out on Friday October 17; due on Friday October 24.
Topics: Compact sets, Heine-Borel theorem, continuous functions on Rn .
Submit for marking problems
Wednesday, November 4 Lecture 22 : Banach contraction principle: Application
Expectations:
1. Apply the Banach Contraction principle when applicable.
22.1 Example Let A = [a ij] be an n n matrix with
Friday, November 13 Lecture 26 : Finite dimensional normed vector spaces.
Expectations:
1. Give a continuous map from n to a finite dimensional normed vector space V
which is 1-1 and onto.
26.1 Recall
Wednesday, November 18 Lecture 28 : Limits of sequences of functions.
Expectations:
1. Define pointwise convergence of a sequence of functions in C(n, m) .
2. Define uniform convergence of a sequence
PMath/AMath 331, Fall 2017 - Assignment 1. Solutions
Posted on Friday, September 15. Due on Friday, September 22 at noon.
Topics: Definition of limit, Properties of limits, Bounded sequences, Squeeze
PMath/AMath 331, Fall 2017 - Assignment 5. Solutions.
Posted on Friday October 13; due on Friday October 20.
Topics: Open and closed subsets of a normed vector space.
Practice problems.
(Not to be sub
PMath/AMath 331, Fall 2017 - Assignment 2. Solutions
Posted on Friday, September 22; due on Friday, September 29 at noon.
Topics: Upper and lower bounds, Least upper bound principle, Monotone converge
PMath/AMath 331, Fall 2017 - Assignment 6. Solutions
Posted on Friday October 20; due on Friday October 27.
Topics: Compact sets, Heine-Borel theorem, continuous functions on Rn .
Practice problems.
(
PMath/AMath 331, Fall 2017 - Assignment 4. Solutions.
Posted on Friday October 6; due on Friday, October 13.
Topics: Abstract normed vector spaces, convergence and completeness in Rn .
Practice proble
PMath/AMath 331, Fall 2017 - Assignment 3. Solutions.
Posted on Friday, September 29; due on Friday, October 6.
Topics: Cauchy sequences, Contractive sequences, inner products, Euclidean norm on Rn .
PMath/AMath 331, Fall 2014 - Assignment 1. Solutions
Handed out on Friday, September 12. Selected questions are due on Friday, September 19
Topics: Denition of limit, Properties of limits, Bounded seq
PMath/AMath 331, Fall 2014 - Assignment 2
Handed out on Friday, September 19; due on Friday, September 26.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergence,
Bolzano We
PMath/AMath 331, Fall 2014 - Assignment 8
Posted on Friday October 31.
This assignment need not be handed in for marking. But it is important do it as if it was.
Solutions will be posted to check your
Friday, October 9 Lecture 12 : Convergence and completeness in a normed vector
space V.
1. Define a convergent sequence in n and in an abstract normed vector space.
2. Show lim n xn = a if and only if
Wednesday, October 7 Lecture 11 : Normed vector spaces II.
Expectations:
1.
2.
3.
4.
5.
Define a norm on a vector space and a normed vector space.
State various examples of norms on n and on C[a, b].
Friday, October 30 Lecture 20 : Contractions and fixed points.
Expectations:
1. Verify in the case of a simple dynamical system (X, T) whether T is a contraction or
not.
20.1 Proposition Suppose S V.
Wednesday, November 11 Lecture 25 : The Intermediate value theorem.
Expectations:
1. State the Intermediate Value Theorem.
2. Recognize that the continuous image of a closed interval is a closed inter
Wednesday, November 25 Lecture 31 : Uniform convergence of series of
functions.
Expectations:
1.
2.
3.
4.
Define an infinite series of functions.
Define a uniformly Cauchy series of functions.
State t
Monday, November 16 Lecture 27 : The generalized Heine-Borel theorem.
Expectations:
1. State the Generalized Heine-Borel theorem for finite dimensional normed vector
spaces.
2. Recognize that, if W is
Friday, November 27 Lecture 32 : Application: Approximating functions with
polynomials.
Expectations:
1. Approximate a function with a polynomial via a Van der Monde matrix.
2. State the Weierstrass a