Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 15, 2009
Assignment 4
1. If f is a continuous mapping of a metric space X into a metric space Y , prove that
f (E ) f (E )
for every set E X . Show, by an example, that f (E ) can be
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: November 3, 2009
Assignment 13
1. Establish the following monotone convergence theorem for the Riemann integral. Assume fk
and f are continuous on I. Then if fk (x)
f (x) for all x I, it follow
Math 653 - Introductory Analysis
Instructor: J. Metcalfe Due: October 27, 2009
Assignment 12 1. (Comp problem) Consider the sequence 11 1 + + + . 23 n Determine whether this sequence converges and justify your answer. an = log n The limit of this sequence
Math 653 - Introductory Analysis
Instructor: J. Metcalfe Due: November 5, 2009
Assignment 14 1. Let p and q be positive real numbers such that 11 + = 1. pq Prove the following statements. (a) If u 0 and v 0, then uv Equality holds if and only if up = v q
Math 653 - Introductory Analysis
Instructor: J. Metcalfe Due: November 24, 2009
Assignment 16 1. Let f L1 (X, ) be given. Show that, for every > 0, there is a > 0 such that S F , (S ) < =
S
|f | d < .
We have |f | M+ , and thus, so is S |f | for any S F .
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: October 20, 2009
Assignment 11
1. (Comp problem) Assume y : R R is a C 1 function satisfying
y (t) ay (t) + b,
t 0,
where a and b are positive constants. Prove that for all t 0 one has
y (t) y
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: October 1, 2009
Assignment 7
1. Prove that the sup norm on C ([0, 1], R) is not derived from an inner product , by f
[Hint: Show that the parallelogram identity fails.]
2
= f, f .
In an inner p
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: October 6, 2009
Assignment 8
1. (Comp problem) For x [1, 1] consider the equation
(1 + sin(f (x)f (x) = x2
(1)
for the unknown function f . Use the Contraction Mapping Theorem to show that ther
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: October 8, 2009
Assignment 9
1. (Comp problem) Let f : Rn Rn be continuously dierentiable. Assume that the n n
matrix Df (x) is invertible for all x Rn , and that
f 1 (K )
is compact, compact K
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: October 13, 2009
Assignment 10
1. (Comp problem) Suppose A Rn is a convex open set and h : A R is dierentiable.
Suppose x, y A and denote the line segment from x to y by L. Prove there is a poi
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 22, 2009
Assignment 6
1. Let F be a set of real-valued, dierentiable functions so that f and f are both continuously
dierentiable. Suppose that for each f F
f (x) = 0
|x| > 1
if
1
(f
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 17, 2009
Assignment 5
1. For n = 1, 2, 3, . . . , x real, put
fn (x) =
x
.
1 + nx2
Show that cfw_fn converges uniformly to a function f , and that the equation
f (x) = lim fn (x)
n
i
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 3, 2009
Assignment 1
1. Prove that a set A M is connected if and only if and A are the only subsets of A that are
open and closed relative to A.
We rst prove = by proving the contrapo
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 8, 2009
Assignment 2
1. Let A X be connected and contain more than one point. Show that every point of A is a
limit point of A.
Suppose not. Let x be a point of A which is not a limit
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: September 10, 2009
Assignment 3
1. Suppose cfw_pn and cfw_qn are Cauchy sequences in a metric space X . Show that the sequence
cfw_d(pn , qn ) converges.
By the triangle inequality
d(pn , qn
Math 653 - Introductory Analysis
Instructor: J. Metcalfe
Due: December 1, 2009
Assignment 17
1. Let f L1 (I, dx). Show that there exist fj C(I) such that fj f almost everywhere.
We showed in class that C(I) was dense in L2 (I, dx). First of all, note that