Math205c Spring Term 2003

Assignment 17

1. Prove continuity of the mapping

K : C([0, 1], ℝ^{n}) → C([0, 1], ℝ^{n}) , f ↦ K(f)

defined through

K(f)(x) := ∫_{0}^{x} f(y) dy , x ∈ [0, 1] .

2. Let (M, d_{M}) be a metric space. Show that

d_{x_{0}} : M → ℝ , x ↦ d_{M}(x, x_{0})

is continuous.

3. Let (M, d_{M}) and (N, d_{N}) be metric spaces and assume that N is complete. Define d_{B(M,N)} through

d_{B(M,N)}(f, g) = sup_{x∈M} d_{N}(f(x), g(x)) , f, g ∈ B(M, N)

and prove that

(B(M, N), d_{B(M,N)})

is complete.

4. Formulate and prove the fact that the composition f ∘ g of continuous functions f, g defined on metric spaces is continuous and give an example where the composition is continuous but nor f or g is continuous.

5. You ask a question.

The Homework is due Friday, April 11.