Book contents
- Frontmatter
- Contents
- Preface
- PART ONE METRIC SPACES
- 1 Calculus Review
- 2 Countable and Uncountable Sets
- 3 Metrics and Norms
- 4 Open Sets and Closed Sets
- 5 Continuity
- 6 Connectedness
- 7 Completeness
- 8 Compactness
- 9 Category
- PART TWO FUNCTION SPACES
- PART THREE LEBESGUE MEASURE AND INTEGRATION
- References
- Symbol Index
- Topic Index
3 - Metrics and Norms
from PART ONE - METRIC SPACES
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- PART ONE METRIC SPACES
- 1 Calculus Review
- 2 Countable and Uncountable Sets
- 3 Metrics and Norms
- 4 Open Sets and Closed Sets
- 5 Continuity
- 6 Connectedness
- 7 Completeness
- 8 Compactness
- 9 Category
- PART TWO FUNCTION SPACES
- PART THREE LEBESGUE MEASURE AND INTEGRATION
- References
- Symbol Index
- Topic Index
Summary
In the beginning there were operations – hundreds of them – limits, derivatives, integrals, sums; all of the many operations on functions, sequences, sets, vectors, matrices, and whatever else you might have encountered in calculus. The hallmark of twentieth-century mathematics is that we now view these operations as functions defined on entire collections of “abstract” objects rather than as specific actions taken on individual objects, one at a time. Maurice Fréchet, in a short expository article from 1950, had this to say (the italics are his own):
In modern times it has been recognized that it is possible to elaborate full mathematical theories dealing with elements of which the nature is not specified, that is, with abstract elements. A collection of these abstract elements will be called an abstract set. If to this set there is added some rule of association of these elements, or some relation between them, the set will be called an abstract space. A natural generalization of function consists in associating with any element x of an abstract set E a number f(x). Functional analysis is the study of such “functionals” f(x). More generally, general analysis is the theory of the transformations y = F[x] of an element x of an abstract set E into an element y of another (or the same) abstract set F. It is obvious that the study of general analysis should be preceded by a discussion of abstract spaces. […]
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- Information
- Real Analysis , pp. 36 - 50Publisher: Cambridge University PressPrint publication year: 2000