Homework set 3 solutions APPM5440, Fall 2006 For problems 1.17, 1.18, 1.20, see attached notes. Problem 1.22: First we prove that if a, b S, then either Ca = Cb , or the two sets are disjoint. Case 1 - suppose that a b. Then c Ca ca cb c Cb . (In the midd

Applied Analysis (APPM 5440): Final Exam
1.30pm 5.00pm, Dec 11, 2005. Closed books.
In proofs, please state clearly what you assume, and what you will prove.
Problem 1: No motivation is required for the following questions: (2p each)
(a) Dene what it mean

Homework set 1 APPM5440, Fall 2006
Problem 2c: Set I = [0, 1] and consider the set X consisting of all continuous
functions on I, with the norm
1
|f | =
|f (x)| dx.
0
Prove that the space X is not complete.
Solution: A straight-forward way of proving this

Applied Analysis (APPM 5440): Midterm 3 Solutions 5.30pm 6.50pm, Dec. 4, 2006. Closed books. Problem 1: No motivation required. 2p each: (a) Let (X, T ) denote a topological space. Specify the axioms that T must satisfy. (b) Let (X, T ) denote a topologic

Applied Analysis (APPM 5440): Midterm 2 5.30pm 6.45pm, Oct. 30, 2006. Closed books. Problem 1: Let (X, d) be a metric space. (a) Dene what it means for a subset of X to be compact. (2p) (b) Let (xn ) be a sequence in . Suppose that there exists an > 0, su

Applied Analysis (APPM 5440): Midterm 3 5.30pm 6.50pm, Dec. 4, 2006. Closed books. Problem 1: No motivation required. 2p each: (a) Let (X, T ) denote a topological space. Specify the axioms that T must satisfy. (b) Let (X, T ) denote a topological space.

Applied Analysis (APPM 5440): Midterm 1 5.30pm 6.45pm, Sep. 25, 2006. Closed books. Problem 1: No motivation required for (a) and (c). Only brief motivations required for (b) and (d). 2 points each: (a) Define what it means for a metric space (X, d) to be

Solutions to homework set 6 - APPM5440, Fall 2006 2.10: Let A denote the set of functions in C(Rn ) that vanish at infinity. That A = Cc follows from the following two claims: Claim 1: Cc is dense in A. Claim 2: A is closed. Proof of Claim 1: Fix an f A.

Homework set 10 APPM5440 Solution sketches Textbook 4.5a: The connected subsets of R are the intervals of the form (a, b), [a, b], (a, b], and [a, b) where a and b are numbers such that a b . A full solution consists of two steps. First, let I denote and

Homework set 8 - APPM5440 - Solutions 3.6: Set X = C([-a, a]) and define on X the operator 1 1 a [F u](x) = u(y) dy + 1. -a 1 + (x - y)2 Then the given equation can be formulated as a fixed point problem u = F (u). We find that |F (u) - F (v)| = sup
x[-a,

Applied Analysis (APPM 5440): Midterm 1 5.30pm 6.45pm, Sep. 25, 2006. Closed books. Problem 1: No motivation required for (a) and (c). Only brief motivations required for (b) and (d). 2 points each: (a) Define what it means for a metric space (X, d) to be

Applied Analysis (APPM 5440): Final Exam 7.30am 10.00am, Dec. 20, 2006. Closed books. Problem 1: The following problems are worth 2p each. No motivation required for (a), (e), or (f). For the rest, please motivate your answer briey (at most one or two sen

Applied Analysis (APPM 5440): Final Exam Solutions 1.30pm 5.00pm, Dec 11, 2005. Closed books. In proofs, please state clearly what you assume, and what you will prove. Problem 1: No motivation is required for the following questions: (2p each) (a) Dene wh

Applied Analysis (APPM 5440): Final Exam 7.30am 10.00am, Dec. 20, 2006. Closed books. Problem 1: The following problems are worth 2p each. No motivation required for (a), (e), or (f). For the rest, please motivate your answer briey (at most one or two sen

Homework set 5 APPM5440
2.4: Lets consider X = [1, 1] instead. Then set f (x) = |x|, and
1 + n x2
fn (x) =
.
n + n2 x2
Then fn f uniformly, fn C (X), and f is not dierentiable. (To justify
the shift we made initially, simply note that if we dene gn C([0,