f (t)) so that f (0

1. hw 8

We have learned several examples of metric spaces in classes. For example, the Euclidean space (R^p, de), the space of bounded functions (B(X), d∞), and the space (l^p(N), dp) of p- summable sequences. In this exercise, we are going to learn some abstract properties of metric spaces. Moreover, we will study continuous functions on abstract metric spaces.

(1) Let (X, d) be a metric space. Let f : R → R be an increasing function (s < t implies f (s) < f (t)) so that f (0) = 0 and

f (s + t) ≤ f (s) + f (t) for any s, t ≥ 0.

Define a function ρ : X × X → R by

df(x, y) = f (d(x, y)), x, y ∈ X.

(a) Show that df defines a (new) metric on X.

(b) For each x, y ∈ X, define

ρ(x, y) = d(x, y) 1 + d(x, y). Show that ρ also defines a metric on X.

(2) Let S be the subset of l²(N) consisting of all functions a : N → R such that a(n) = 0 for all but finitely many n ∈ N. We define a function f : S → l²(N) by

f (a₁, a₂, · · · ) = (a₁,√ 2a₂,√

3a₃, · · · ).

Prove or disprove that f is continuous at 0. Here 0 = (0, 0, · · · ).

(3) It is known that C([0, 1]) is a vector subspace of B([0, 1]). Prove that C([0, 1]) is a closed subset of (B([0, 1]), d∞). (This implies that (C([0, 1], R), k · k∞) is also a Banach space.)

(4) Let r > 0 and D(0, r) = {x ∈ Rⁿ: kxk ≤ r}. Here k · k denotes the Euclidean norm on Rⁿ. Suppose that

f : D(0, r) → Rⁿ is a map satisfying

(a) kf (0)k ≤ r/3

(b) kf (x) − f (y)k ≤ 2kx − yk/3 for x, y ∈ D(0, r).

Show that there exists x₀ ∈ D(0, r) so that f(x0) = x₀. Hint: use contraction mapping principle.

(5) The following integral equation for f : [−1, 1] → R arises in a model of a motion of gas particle on a line:

f (x) = 1 + 1 π

Z 1

−1

1

1 + (x − y)²f (y)dy, −1 ≤ x ≤ 1.

Prove that f is a unique solution in C[−1, 1]. Hint: use contraction mapping princi- ple.

(6) Prove that the following differential equation

y⁰(x) = sin (xy(x)) , y(0) = 0

has a unique solution on C[0, 1] and solve for y(x). Hint: use contraction mapping principle. Consider the map

T : C[0, 1] → C[0, 1], T (f )(x) = Z x

0

sin(tf (t))dt.

1

2

Prove that T is a contraction. Prove that f (t) is a solution for the differential equation if and only if T (f ) = f.