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Unformatted text preview: formula for h(Cr , Cs ). Prob 2.2.7 Completing the Hilbert space-flling curve example. (a) For functions f, g : [0, 1] → [0, 1]2 , the sup metric is ρ(f, g ) = sup{d(f (t), g (t)) : 0 ≤ t ≤ 1} where d(f (t), g (t)) is the Euclidean distance between the points f (t) nad g (t). Show ρ is a metric. (b) A sequence f1 , f2 , f3 , . . . is Cauchy if for every ǫ > 0 there is an N for which i, j > N implies ρ(fi , fj ) < ǫ. Show the sequence of curves C0 , C1 , C2 , . . . of example 2.2.3 is a Cauchy sequence in the sup metric. (c) Find a reference that the space of continuous functions [0, 1] → [0, 1]2 , with the sup metric, is complete. That is, every Cauchy sequence of elements of this space converges to an element of this space. If you’re ambitious and have some time, try to prove this yourself. (d) Use the fact that the continuous image of a closed set is closed to deduce the limit curve C∞ has image [0, 1]2 , and so is spacefilling. 44 2.3 CHAPTER 2. ITERATED FUNCTION SYSTEMS Convergence of deterministic IFS The...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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