MA222 2015-2016 Problem Set 1 PDF

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METRIC SPACES MA222 — 2015/2016 Problems 1 1. For 0 < p < 1 show that dp (x, y) = (|x1 − y1 |p + |x2 − y2 |p )1/p is not a metric on R2 . 2. The distance of two nonempty subsets A, B of a metric space M is defined as d(A, B) = inf x∈A,y∈B d(x, y). For each axiom of metric, either show that it holds for distance of sets or give a counter example. 3. Suppose that M is a metric space with metric d and let f : M → R. Show that d1 (x, y) = d(x, y) + |f (x) − f (y)| is a metric on M . How do the balls with respect to this metric look like in the special case when M = R and f (x) = 0 for x ∈ Q and f (x) = 1 for x ∈ / Q? 4. If d is a metric on M, show that min(1, d(x, y)) is also a metric on M . Compare the balls in these two metrics. 5. Notice that the requirement that d be positive has three parts: d(x, y) ≥ 0, d(x, x) = 0 and d(x, y) = 0 =⇒ x = y. Often we encounter situations where the last part is not satisfied; in that case we speak about a pseudometric. (The symmetry and triangle inequality properties are still assumed to hold.) Show that if d is a pseudometric, then the relation x ∼ y ⇐⇒ d(x, y) = 0 is an equivalence relation and d, naturally defined on the equivalence classes, becomes a metric. 6. Suppose that d : M × M → R satisfies d(x, y) = 0 iff x = y, and d(x, z) ≤ d(x, y) + d(z, y). Show that d is a metric on M . 7. Sketch various balls in the “jungle river metric in R2 ” ( |x2 − y2 | if x1 = y1 d(x, y) = |x2 | + |y2 | + |x1 − y1 | otherwise p 8. Given λ1 , λ2 ∈ R, define kxk = |λ1 x21 + λ2 x22| for x = (x1 , x2 ) ∈ R2 . For what λ1 , λ2 does k · k satisfy the triangle inequality? n n |x |. Show that kxk 9. For x ∈ Rn , let kxk∞ = maxi=1 i ∞ is a norm in R . Pn |xi − yi |p is a metric on Rn . 10⋆ For 0 < p < 1 show that dp (x, y) = i=1...