MTH6127 - Metric Spaces and Topology

Assignment Task

1. For the metric d_L1 (f, g) defined by d_L1 (f, g) = b |f (x) − g(x)|dx, a where f, g ∈ C[a, b], compute the distance d_L1 (f, g) between f (x) = e and g(x) = 2 where [a, b] = [0, 5].

2. Let X = R^m. For any x = (x₁, ..., x_m), y = (y₁, ..., y_m) ∈ X, we set d_∞(x, y) := max{|x_k − y_k|}. k Prove that d_∞ defines a metric on X.

3. Let (X, d) be a metric Define two new functions d_a and d_b on X × X by d(x, y) d_a(x, y) := min{d(x, y), 1}, d_b(x, y) := 1 + d(x, y), for x, y ∈ X. Prove that d_a and d_b are also metrics on X.

4. We define “the Jungle metric” d_J on X = R² by  |x₂ − y₂| if x₁ = y₁, d_J (x, y) :=  |x₂| + |x₁ − y₁| + |y₂| otherwise. (“climb down from the tree, walk to another one, climb up the tree”). Prove that d_J defines a metric on X.

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