Functional Analysis pp 19-62 | Cite as

# Introduction to Metric Spaces

Chapter

## Abstract

We have defined the symbols ℝ and ℂ as the sets of real and complex numbers, respectively. We can specify the position of a point in three-dimensional space by its coordinates (

*x*_{1},*x*_{2},*x*_{3})*x*_{i}∈ ℝ in some Euclidean frame. We write*x*= (*x*_{1}*x*_{2}*x*_{3}) and say*x*is in ℝ^{3}, which we write*x*y∈ ℝ^{3}.The*Euclidean distance*between two points*x, y*∈ ℝ^{3}is$$ {{d}_{E}}(x,y) = {{\left\{ {\sum\limits_{{i = 1}}^{3} {{{{({{x}_{i}} - {{y}_{i}})}}^{2}}} } \right\}}^{{{{1} \left/ {2} \right.}}}}. $$

## Keywords

Equivalence Class Linear Space Limit Point Product Space Cauchy Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

## The classical text on functional analysis is

- F. Riesz and B. Sz.-Nagy,
*Functional Analysis*, Frederick Ungar Publishing Co., New York, 1955.Google Scholar

## Among the many other excellent treatises we mention

- A.N. Kolmogorov and S.V. Fomin,
*Introductory Real Analysis*, Dover Publications Inc., New York, 1975. This has extensive discussion on set theory, on measure theory and integration.Google Scholar - A. Friedman,
*Foundation of Modern Analysis*. Dover Publications Inc., New York, 1970. This covers much of the material in our book at greater depth and level of abstraction. In particular it has an extensive study of Lebesgue integration, and of the concept of the adjoint for spaces other than Hilbert spaces.Google Scholar - L.V. Kantorovich and G.P. Akilov,
*Functional Analysis*, Pergamon Press, 1982. This is an extensive work with copious references to the original literature and to other treatises.zbMATHGoogle Scholar

## A comprehensive treatment of functional analysis at an abstract level may be found in

- K. Yosida,
*Functional Analysis*, Springer-Verlag, New York, 1971.zbMATHGoogle Scholar

## A brief, easily readable account of some aspects of functional analysis may be found in

- C.W. Groetsch,
*Elements of Applicable Functional Analysis*, Marcel Dekker, New York, 1980.zbMATHGoogle Scholar

## An exemplary text book which covers much of the material in the present book at much greater depth, and which has many examples and references is

- E. Kreyszig,
*Introductory Functional Analysis with Applications*. Robert E. Krieger Publishing Company, Malabar, Florida, 1989.zbMATHGoogle Scholar

## Copyright information

© Kluwer Academic Publishers 1996