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11

Metric spaces and normed spaces

11.1 Metric spaces: examples

In Volume I, we established properties of real analysis, starting from the properties of the ordered field R of real numbers. Although the fundamental properties of R depend upon the order structure of R, most of the ideas and results of the real analysis that we considered (such as the limit of a sequence, or the continuity of a function) can be expressed in terms of the distance d(x, y) = |xy| defined in Section 3.1. The concept of distance occurs in many other areas of analysis, and this is what we now investigate.

A metric space is a pair (X, d), where X is a set and d is a function from the product X × X to the set R+ of non-negative real numbers, which satisfies

1.d(x, ...

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