Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Browse subjects
  3. Mathematics
  4. Real and Complex Analysis
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

3279 results in Real and Complex Analysis

Page 30 of 164

  • 5 - Metric and Euclidean Spaces

  • John B. Conway, George Washington University, Washington DC
  • Book:
    A First Course in Analysis
    Published online:
    28 May 2018
    Print publication:
    25 July 2017, pp 115-154

  • Summary

    In this chapter we begin the study of p-dimensional Euclidean space, Rp, but in this beginning we will carry it a step further. We want to discuss differentiation and integration on Rp, but first we need to extend the notions of sequential convergence, the properties of sets, and the concept of continuity to the higher dimensional spaces. The effort to explore these concepts in Rp, however, is not greater than what is required to explore these notions in what are called metric spaces. In many respects the abstract spaces are easier to cope with than Rp. Moreover some of what we have already done is properly couched in metric spaces. Indeed, the material of §4.1 can be set there with little additional effort. Nevertheless during this venture into abstraction the main set of examples will be Euclidean space.

    We start with the concept of distance between points. This must be general enough to encompass a variety of circumstances, but it should conform to the intuitive notion we all have of what is meant by distance. Since this is done at the start of the first section, it would be profitable before proceeding for the reader to reflect on what properties (s)he thinks should be included in an abstract concept of distance; then you can compare your thoughts with the definition that starts the following section.

    The treatment of metric spaces here is based on Chapter 1 of [4].

    Definitions and Examples

    Definition. A metric space is a pair of objects, (X, d), where X is a set and d is a function d : X × X → [0,∞) called a metric, that satisfies the following for all x, y, z in X:

  • (a) d(x, y) = d(y, x);

  • (b) d(x, y) = 0 if and only if x = y; and

  • (c) (Triangle Inequality) d(x, y) ≤ d(x, z) + d(z, y).

    • Condition (a) is sometimes called the symmetric property and says that the distance from x to y is the same as the distance from y to x. The second property says the obvious: the distance from a point to itself is 0 and the only point at a distance zero from x is x itself.

Page 30 of 164