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Assignment Task

1. For the metric dr1 (f,g) defined by dr1 (f, 9) = [ \f (x) − g(x)\dr, where f, gЄ C[a, b], compute the distance dr1(f, g) between f(x) = e and g(x) = 2 where [a, b] [0,5].
2. Let X = Rm. For any = (1,Im), Y = (y1,Ym) EX, we set d(x, y) = max{|k − yk|}. – Prove that do defines a metric on X.
3. Let (X, d) be a metric space. Define two new functions do and do on X x X by da(x, y) = min{d(x, y), 1}, d(x, y) := = d(x, y) 1+d(x, y)’ for x, y Є X. Prove that do and do are also metrics on X.
4. We define “the Jungle metric” dy on X = R2 by |X2 Y2 dj (x,y) if x1 = y1,|x2|+|1-y1|+|y2| otherwise. (“climb down from the tree, walk to another one, climb up the tree”). Prove. dj defines a metric on X.