Math205b Winter Term 2003

Assignment 15

1. Let α ∈ (0, 1) and assume that A ⊂ C^{α}([0, 1], 𝕂) be a bounded subset, that is, assume that

∥f∥_{α} ≤ M , f ∈ A

for some M > 0. Show that A is uniformly equicontinuous.

2. Let f, g ∈ C_{c}(ℝ, 𝕂) and show that

supp(f ∗ g) ⊂ supp(f) + supp(g)

=: {x + y | x ∈ supp(f) , y ∈ supp(g)} .

3. Let 0 ≤ f ∈ C_{c}(ℝ, ℝ) with ∫_{−∞}^{∞} f(x) dx = 1. Show that (f_{n})_{n∈ℕ} defined through

f_{n}(x) = n f(n x) , x ∈ ℝ , n ∈ ℕ ,

is an approximate identity. Let (M, d) be a metric space. The set

𝔹(x, ε) = {y ∈ M | d(x, y) < ε}

is called “open” ball of radius ε > 0 about x ∈ M. A set O ⊂ M is called open iff ∀x ∈ O ∃ ε > 0 s.t. 𝔹(x, ε) ⊂ O .

4. Let (M, d) be a metric space and show that

τ := {O ⊂ M | O is open}

defines a topology on M. Compute τ for M = ℝ and d defined by

d(x, y) = \begin{cases}0 , & x = y \\ 1 , & x ≠ y . \end{cases}

For M = ℝ define

d(x, y) = \frac{2}{π} ∫_{min(x,y)}^{max(x,y)} \frac{dξ}{1 + ξ^{2}} , x, y ∈ ℝ .

Show that (M, d) is a metric space and compute 𝔹(0, 1), 𝔹(1, \frac{π}{4} ) .

5. You ask a question.

The Homework is due on Friday, March 7