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.
Problem 1. Review denitions 28.1, 28.2 and theorem 28.3. Co
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 functions.
Problem 1. (a) Show that, if f : Rn Rm and g : Rn
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 lim n | xn a | = 0.
3. Define Cauchy sequence in n , c
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].
Be able to verify that certain simple functions on a ve
Monday, October 5 Lecture 10 : Normed vector spaces I.
Expectations:
1. Define a norm on a vector space and a normed vector space.
2. State various examples of norms on n and on C[a, b].
3. Be able to verify that certain simple functions on a vector space
Wednesday, October 14 Lecture 13 : Closed sets in a normed vector space.
Expectations:
Define limit point in V .
Define closed subset of V .
Show that a subset of n is closed or not closed.
Recognize and apply the fact that finite unions of closed sets ar
Monday, October 19 Lecture 15 : Compact sets in normed vector spaces.
Expectations:
1.
2.
3.
4.
Define a compact set.
Define a bounded subset of a normed vector space V.
Show a set is compact or is not compact from the definition.
Show that closed subsets
Friday, October 23 Lecture 17 : Continuous functions mapping V1 V2
Expectations:
1. Define the limit of a function f : V1 V2.
2. Define continuity of a function f : V1 V2 at a point a.
3. State the sequential definition of a continuous function at a point
Wednesday, October 21 Lecture 16 : The Heine-Borel theorem.
Expectations:
1. State the Heine-Borel theorem.
2. Apply the Heine-Borel theorem in various problems.
3. State and apply the Cantor intersection theorem.
We will now show that, in n, the compact
Friday, October 16 Lecture 14 : Open sets in a normed vector space.
Expectations:
1.
2.
3.
4.
5.
6.
Define an open ball of radius r in V.
Define an open subset in V.
Recognize when certain subsets of V are open or not open.
Characterize an open set by its
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 n.
Use the Euclidean norm to find the distance between
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 Definition The sequence cfw_xn of real numbers is said to be
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 determined that a contractive sequence cfw_xn converges
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, uniform
convergence of functions.
Problem 1. Review denitio
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 Bolzano-Weierstrass theorem.
Apply the Bolzano-Weierstrass th
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 limit of a sequence in , when it exists, is unique.
State t
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 the Squeeze theorem to show a sequence converges.
2.0 In
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 infinite countable and uncountable sets.
Describe gene
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 Cauchy convergence criterion).
Use the Cauchy converge
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 .
State the Least Upper Bound Principle.
State the Monot
Monday, October 26 Lecture 18 : Some continuous functions: Linear
transformations and Lipschitz functions.
Expectations:
1.
2.
3.
4.
5.
Define a Lipschitz function on a subset of V1.
Prove that Lipschitz functions are continuous functions on their domain.
Solution 5: AMATH/PMATH 331, 2015
1. D is a subset of the normed vector space C[0, 2] with the uniform norm. fk 1. Thus the set D is
bounded. As proved in Problem 6 of Assignment 4, the sequence cfw_fk converges pointwise to the following
function:
0 fo
PMath/AMath 331, Fall 2015 - Assignment 5.
Posted on Friday October 16; due on Friday October 23.
Topics: Open and closed subsets of a normed vector space.
Practice problems.
(Not to be submitted for marking. These are to help students learn and better un
PMath/AMath 331, Fall 2015 - Assignment 4.
Posted on Friday October 9; due on Friday, October 16.
Topics: Abstract normed vector spaces, convergence and completeness in Rn .
Practice problems. (Not to be submitted for marking. These are to help students l
PMath/AMath 331, Fall 2015 - Assignment 5. Solutions.
Posted on Friday October 16; due on Friday October 23.
Topics: Open and closed subsets of a normed vector space.
Practice problems.
(Not to be submitted for marking. These are to help students learn an
PMath/AMath 331, Fall 2015 - Assignment 6.
Posted on Friday October 23; due on Friday October 30.
Topics: Compact sets, Heine-Borel theorem, continuous functions on Rn .
Practice problems.
(Not to be submitted for marking. These question are to help stude
PMath/AMath 331, Fall 2015 - Assignment 7.
Posted on Friday October 30.
Topics: Lipschitz functions.
Practice problems.
Problem P1. Find a bounded continuous function on some subset S of R that is not
Lipschitz on S. HINT: The derivative should become arb
PMath/AMath 331, Fall 2015 - Assignment 6. Solutions
Posted on Friday October 23; due on Friday October 30.
Topics: Compact sets, Heine-Borel theorem, continuous functions on Rn .
Practice problems.
(Not to be submitted for marking. These question are to
Topics to review for AMPM331 midterm, Fall 2015.
1) The real numbers .
Countable and uncountable subsets of .
Convergent sequences in .
Basic limit properties.
Bounded and unbounded subsets of .
Supremum and infimum of a subset of .
The Least upper
Solution 2: AMATH/PMATH 331, 2016
1. Note that
n
3n + 3n = 3 2.
Since n 2 1 as n , we get using the squeeze theorem n 2n + 3n 3 as n .
2. Note that a1 = 0, a2 = 5 + 2a1 =
Assume an
an+1 4. Then 2an 2an+1 8, and
5 4.
5 + 2an 5 + 2an+1 13. Hence 5 + 2an
Solution 4: AMATH/PMATH 331, 2016
1. Assume that cfw_xk converges in V . Then cfw_xk is a Cauchy sequence. Thus for any > 0, N > 0 such that
for all k, m > N , kxk xm k < . By the triangle inequality, we have
kxk k kxk xm k + |xm k, kxm k kxm xk k + kxk
Wednesday, September 21 Lecture 6 : Cauchy sequences.
Concepts: Define a Cauchy sequence in , prove that every convergent sequence in is
Cauchy, define a complete subset of , state the Completeness theorem in (The Cauchy
convergence criterion), use the Ca
Friday, September 9 Lecture 1 : The Real numbers.
Concepts: Definition of the real numbers, density of the reals, countable sets and uncountable sets,
examples of infinite countable and uncountable sets, Cantor set K.
Most of the content of Lecture 1 is l