Math205c Spring Term 2003

Assignment 24

1. Let (M, d_{M}) and (N, d_{N}) be metric spaces. Show that

f ∈ C(M, N) ⟺ f(\overline{A}) ⊂ \overline{f(A)} , A ⊂ M .

2. Let f : (−1, 1) → ℝ be such that

f(0) = 0 and f(x) ≥ c |x|^{α} , x ∈ (−1, 1)

for some α ∈ (0, 1) and some c > 0. Conclude that f is not differentiable at x = 0.

3. Show that O(n) := {M ∈ ℝ^{n×n} | MM^{T} = 1_{n}} is a compact subset of ℝ^{n×n}.

4. Show that the series

∑_{n=1}^{∞}(1 − cos(x/n))

converges uniformly on compact subsets of ℝ.

The Homework is due Friday, May 30.