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Metric Spaces, Lecture Notes - Mathematics

Calculus

Post: October 8th, 2011
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Metric spaces, Metrics a Nation of Distance, Examples, Convergence
Metric spaces, Metrics a Nation of Distance, Examples, Convergence
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Metric Spaces Adrian Down 16779577 August 2, 2005 1 Metrics: a notion of distance So far in analysis on R, every definition (convergence, Cauchy, continuity) has included statements like “|x − y| < ”. The idea is that this |x − y| represents a distance from x to y. A distance should satisfy the properties: i) |x − y| ≥ 0, ∀x, y ∈ R ii) |x − y| = 0 → x = y iii) |x − y| = |y − x| iv) “Triangle inequality”: |x − y| + |y − z| ≥ |x − z|, ∀x, y, z Definition: a metric space is a set of “objects” S together with a function d on the set of pairs (x, y), x, y ∈ S such that d(x, y) ∈ R, ∀x, y ∈ S 1)d(x, y) ≥ 0, ∀x, y ∈ S 2)d(x, y) = 0 → x = y 3)d(x, y) = d(y, x), ∀x, y ∈ S 4)d(x, y) + d(y, z) ≥ d(x, z), ∀x, y, z ∈ S d is called the metric on S. May write (S, d) to specify the metric. 4) is called the triangle inequality; it is the most important, and it is usually the hardest to verify. 2 Examples 1) (R, d) is a metric s..

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