Quiz, week 7

  1. Suppose the bar menu offers A⊆XA \subseteq X glasses of wine, inside a metric space (X,d₂)(X, d₂), where X⊆ℝX \subseteq \mathbb{R}. Suppose your utility function is u(x)=xu(x) = x. Give examples of where there is no optimal choice, because:
    1. The menu AA is not a closed set.
    2. The menu AA is a closed set, but the metric space is not complete.
  2. What are the fixed points of:
    1. f(x)=xf(x) = x where f:[0,1]→[0,1]f : [0, 1] → [0, 1].
    2. f(x)=1−xf(x) = 1 - x where f:[0,1]→[0,1]f : [0, 1] → [0, 1].
    3. f(x)=x²f(x) = x² where f:[0,1]→[0,1]f : [0, 1] → [0, 1].
  3. Which of these functions are self-maps?
    1. f(x)=xf(x) = x where f:[0,1]→[0,1]f : [0, 1] → [0, 1].
    2. f(x)=x+1f(x) = x + 1 where f:[0,1]→[0,2]f : [0, 1] → [0, 2].
    3. f(x)=x+1f(x) = x + 1 where f:ℝ→ℝf : \mathbb{R} → \mathbb{R}.
    4. T(f)=f(0)T(f) = f(0) where T:CB[0,1]→ℝT : CB[0, 1] → \mathbb{R}. e.g. If f(x)=1−xf(x) = 1-x, then T(f)=f(0)=1T(f) = f(0) = 1.
    5. T(f)=x↦f(x)²T(f) = x \mapsto f(x)² where T:CB[0,1]→CB[0,1]T : CB[0, 1] → CB[0, 1]. e.g. If f(x)=1−xf(x) = 1-x, then T(f)=x↦(1−x)²T(f) = x \mapsto (1-x)².
  4. Which of these spaces are complete?
    1. (ℝ²,d₂)(\mathbb{R}², d₂)
    2. ([0,1],d₂)([0, 1], d₂)
    3. ([0,1),d₂)([0, 1), d₂)
    4. (CB[0,1],d∞)(CB[0, 1], d_∞)
    5. (CB[0,1],d₁)(CB[0, 1], d₁), where d₁(f,g)=∫01|f(x)−g(x)|dxd₁(f, g) = ∫_0^1 |f(x) - g(x)| \, dx