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*) = |*x* – *y*| 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*, ...

Start Free Trial

No credit card required