MTH-533 ASSIGNMENT 1
LEO T. BUTLER
A. Text: 8.1.1d, 8.1.4b, 8.1.6c
p
p
1
B. Let |x|p = (|x1 | + + |xn | ) p denote the p-norm of the vector x Rn , p 1. Let
p
Br = x Rn s.t. |x|p r .
2
(a) Is there a c > 0 such that B 2 B1 Bc ? If so, what is the smallest
MTH-533 (22243170) TUTORIAL 4
LEO T. BUTLER
A. Let E Rn be a closed set. Show that there is a continuous function f : Rn [0, ) R such
that f 1 (0) = E .
Date : January 9, 2014.
1
MTH-533 (22243170) TUTORIAL 3
LEO T. BUTLER
A. Find the interior, closure and boundary of each of the following sets:
(a)
1
| n Z, n = 0 .
E=
n
(b)
E=
nZ
n>0
1
1
,
n+1 n
.
B. Let N be a norm on Rn . Prove that N is equivalent to the 1-norm. That is, there
MTH-533 TUTORIAL 2
LEO T. BUTLER
A. In this problem, we will prove the triangle inequality holds for a p-norm.
(a) Let p > 1 and dene the conjugate of p to be q where 1/p + 1/q = 1. Clearly, q is the conjugate
of p i p is the conjugate of q . (Note that w
MTH-533 TUTORIAL 1
LEO T. BUTLER
A. Text: 8.2.3, 8.2.5d
B. Let A L(R2 ; R2 ) be a linear transformation that preserves the 2-norm (Euclidean length) of every
vector in R2 . Show that
cos t sin t
cos t
sin t
A=
or
sin t cos t
sin t cos t
for some t.
C. Pro
MTH-533 (22243170) ASSIGNMENT 8
LEO T. BUTLER
A. Let E Rn and f : E R be a continuous function. Recall that a E is a local maximum
o
point for f if r > 0 such that x Br (a) E implies f (x) f (a); a E is a global maximum
point if f (x) f (a) for all x E .
MTH-533 (22243170) ASSIGNMENT 7
LEO T. BUTLER
A. Text: 11.4.1
B. Text: 11.4.3
C. Let u : R2 R be a twice dierentiable function. The 2-dimensional wave equation is
2u
2u
= 2,
x2
t
2
where (x, t) are coordinates on R . Prove that if f, g : R R are twice die
MTH-533 (22243170) ASSIGNMENT 6
LEO T. BUTLER
A. Prove that a locally constant function (see assignment 4) is continuous. Deduce that a locally constant
function on a connected domain is constant.
B. Text: 11.2.4.
C. Text: 11.2.7
D. Text: 11.2.8
E. Text:
MTH-533 (22243170) ASSIGNMENT 5
LEO T. BUTLER
A. Theorem 9.15v (page 315) is false. Construct an example to show this.
B. (See Tutorial #4). Let E Rn be a closed, non-empty set and N : Rn R be a norm. Prove that
the function
f (x) = inf cfw_N (x a) s.t. a
MTH-533 (22243170) ASSIGNMENT 4
LEO T. BUTLER
A. Text: 8.4.8b, 9.2.1, 9.2.4, 9.3.3a
B. Say that a function f : A Rn R is locally constant, if for each x A, there is an open set
U x such that f |A U is constant. Prove that if A is connected, then f is loca
MTH-533 (22243170) ASSIGNMENT 2
LEO T. BUTLER
A. Text: 8.3.1ace, 8.3.2
A. Answer the following.
(a) Let f : R R be a real-valued function. State the
point x = a.
denition of continuity of f at the
(b) Prove that every bounded open (resp. closed) interval
MTH-533 (22243170) TUTORIAL 5
LEO T. BUTLER
A. Prove the intermediate value theorem: If E Rn is connected and f : E R is continuous, then,
given a, b f (E ) where a < b, and c [a, b], there is an x E such that f (x) = c.
10
8
b
6
4
y
c
2
0
2
a
4
6
0
1
2
3