201A nal practice - Thomases
1. Let f : [0, 1] (0, 1) be a function of class C 1 - that is to say both f
and f are continuous functions. Suppose further that
max |f (x)| 1
0x1
for some
> 0.
(a) Show that f has exactly one xed point.
(b) Let x0 denote any
201A, Fall 10, Thomases
Homework 3
1. Let X be a normed linear space. A series
xn in X is absolutely
convergent if
xn converges to a nite value in R. Prove X is a
Banach space if and only if every absolutely convergent series converges.
2. Let p : R2 R be
201A, Fall 10, Thomases
Homework 2
1. If f is a function, and S is a bounded subset of R, then the following
statements are equivalent:
(i)f is uniformly continuous on S
(ii) if cfw_sn is a Cauchy sequence in S then cfw_f (sn ) is a Cauchy sequence.
n=1
201A, Fall 10, Thomases
Homework 5
1. If f is continuous on [0, 1] and if
1
f (x)xn dx = 0,
n > 0,
0
then f (x) = 0 on [0, 1].
Hint. The integral of the product of f with any polynomial is zero.
1
Use the Weierstrass theorem to show that 0 f 2 (x) dx = 0.
201A, Fall 10, Thomases
Homework 6
Due on November 10th, 2010
1. For any f C ([0, 1]), we dene
1/2
1
f
1
2
|f (x)| dx
=
0
and
1/2
1
f
(1 + x)|f (x)|2 dx
=
2
.
0
Show that 1 and
that these are norms.
2
are equivalent norms in C ([0, 1]). Note, rst show
2.
201A, Fall 10, Thomases
Homework 8
Due December 1st , 2010
1. Prove that a closed convex subset of a Hilbert space has a unique minimum norm.
2. Prove that a normed linear space X is an inner product space with a norm derived from the
inner product if and
201A, Fall 10, Thomases
Homework 7
Due on November 19th, 2010
1. Suppose cfw_xn n is a weakly convergent sequence in a Banach space X .
Show that the weak limit of cfw_xn is unique.
2. Find the kernel and the range of the linear operator K : C ([0, 1])
C
201A, Fall 10, Thomases
Homework 1
1. Let (X, d) be a metric space. Suppose that cfw_xn X is a sequence
n=1
and set n := d(xn , xn+1 ). Show that for m > n
m1
d(xn , xm )
k
k=n
k .
k=n
Conclude from this that if
d(xn , xn+1 ) <
k =
k=1
n=1
then cfw_xn
201A, Fall 10, Thomases
Homework 7
Mihaela Ifrim
1. Suppose cfw_xn is a weakly convergent sequence in a Banach space X . Show
that the weak limit of cfw_xn is unique.
Proof. Suppose cfw_xn converges weakly to x and y in X . Then for any
X , we nd
lim (xn
201A, Fall 10, Thomases
Mihaela Ifrim
Homework 6
Due on November 10th, 2010
1. For any f C ([0, 1]), we dene
1/2
1
f
1
2
|f (x)| dx
=
0
and
1/2
1
f
2
(1 + x)|f (x)|2 dx
=
.
0
Show that
1
and
2
are equivalent norms in C ([0, 1]).
Proof.
Let w(x) > 0 be
201A, Fall 10, Thomases
Homework 8
Mihaela Ifrim
1. Prove that a closed convex subset of a Hilbert Space has a unique minimum norm.
Proof. Suppose K is a closed convex set in a Hilbert space H. Let
d = inf x .
xK
If d = 0, then we can nd a sequence (xn )
201A Final Exam -December 9, 2010 - Thomases
Solutions
Name:
Show all of your work, in particular note any theorems you may use and
state the conditions carefully and make sure they are satised.
Problem Possible Points Points Received
1
10
2
10
3
10
4
10
201A nal practice solutions - Thomases
1. Let f : [0, 1] (0, 1) be a function of class C 1 - that is to say both f
and f are continuous functions. Suppose further that
max |f (x)| 1
0x1
for some
> 0.
(a) Show that f has exactly one xed point.
(b) Let x0
201A, Fall 10, Thomases
Homework 4
1. Suppose fn C ([0, 1]) is a monotone decreasing sequence that converges pointwise to f C ([0, 1]). Prove that fn converges uniformly
to f. This result is called Dinis monotone convergence theorem.
2. Consider the space