Math Analysis..

Math-4301 Mathematical Analysis I Homework 4

Due on: November 5, 2015 Name:

Instructions. Please solve the following problems (show all your work). You can use your textbooks, class notes. Work on your own and do not discuss the problems with your classmates or anyone else. Please let me know if you have any questions concerning the problems or you do need some hints.

1. (20 pts) Consider the space S = R2 equipped with the metric ed : R2 �R2 ! R de�ned by ed(x;y) = kx�yk

1+kx�yk ; where x; y 2 R2 and kxk =

q x21 +x

2 2; x = (x1;x2):

For " > 0 describe (as a set in R2 using the Cartesian coordinates) the open ball (with respect to the metric ed) eB(x;�), eB(x;�) = fx0 2 R2 : ed(x;x0) < ":g (Hint: Consider two cases: 1 > " > 0 and " � 1.)

2. (20pts) A set A in a metric space (S; d) is called bounded if and only if

9R > 0 3 A � B(x0; R);

for some x0 2 S. Put S = R2 and consider two metric spaces (S; d2) �the space R2 equipped with the usual Euclidean metric:

d2(x;y) = p (x1 �y1)2 +(x2 �y2)2; x = (x1;x2); y = (y1;y2) 2 R2;

and (S;dh) �the space R2 equipped with the highway metric:

dh(x;y) :=

( jx2 �y2j if x1 = y1; jx2j+ jx1 �y1j+ jy2j if x1 6= y1:

Prove that a set A � S is bounded in (S; d2) if and only if it is bounded in (S; dh)

3. (20 pts) Let (V; k k) be a normed space.

a) Show that a closure of an open ball in V is a closed ball (of the same radius), i.e.

B(x0; r) = fx 2 V j kx0 �xk � rg ;

where x0 2 V and r > 0.

b) Show that if (V; k k) is complete and A � V is closed, then (A; k k) is complete.

4. (20 pts) Let (X; d) be a metric space and f1;f2; : : : ;fn : X ! R continuous functions.

a) Show that the function f : X ! R, de�ned by

f(x) = maxff1(x); f2(x); : : : ; fn(x)g ; x 2 X

is continuous.

b) If the functions f1; f2; : : : ; fn are uniformly continuous, does the function f is also uniformly continuous?

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c) Give an example of a sequence of continuous functions fk : X ! [0; 1], k 2 N, such that

f(x) = supffk(x) j k 2 Ng ; x 2 X;

is not continuous.

5. (20 pts) Let (X; d) be a metric space; let A be a nonempty subset of X. For each x 2 X we de�ne the distance from x to A by equation:

dist(x; A) = inf fd(x;a) j a 2 Ag

Let f : X ! R be the function de�ned as follows: f(x) = d(x; A), for all x 2 X.

a) Show that f is continuous.

b) Show that x 2 A if and only if f (x) = 0:

c) Show that every closed subset A � X is the intersection of open sets in X.

d) Show that if A is compact, then f (x) = d(x;a), for some a 2 A.

e) De�ne the �-neighborhood of A in X to be the set

U (A; �) = fx 2 X j d(x;A) < �g :

Show that U (A; �) equals the union of the open balls Bd (a; �), for a 2 A.

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