Continuous Functions of Vector Variables
This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding con...
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| Format: | eBook |
| Language: | English |
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Boston, MA
Birkhäuser Boston
2002, 2002
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| Edition: | 1st ed. 2002 |
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| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Euclidean Space
- 1.1 Multiple Variables
- 1.2 Points and Lines in a Vector Space
- 1.3 Inner Products and the Geometry of Rn
- 1.4 Norms and the Definition of Euclidean Space
- 1.5 Metrics
- 1.6 Infinite-Dimensional Spaces
- 2 Sequences in Normed Spaces
- 2.1 Neighborhoods in a Normed Space
- 2.2 Sequences and Convergence
- 2.3 Convergence in Euclidean Space
- 2.4 Convergence in an Infinite-Dimensional Space
- 3 Limits and Continuity in Normed Spaces
- 3.1 Vector-Valued Functions in Euclidean Space
- 3.2 Limits of Functions in Normed Spaces
- 3.3 Finite Limits
- 3.4 Continuity
- 3.5 Continuity in Infinite-Dimensional Spaces
- 4 Characteristics of Continuous Functions
- 4.1 Continuous Functions on Boxes in Euclidean Space
- 4.2 Continuous Functions on Bounded Closed Subsets of Euclidean Space
- 4.3 Extreme Values and Sequentially Compact Sets
- 4.4 Continuous Functions and Open Sets
- 4.5 Continuous Functions on Connected Sets
- 4.6 Finite-Dimensional Subspaces of Normed Linear Spaces
- 5 Topology in Normed Spaces
- 5.1 Connected Sets
- 5.2 Open Sets
- 5.3 Closed Sets
- 5.4 Interior, Boundary, and Closure
- 5.5 Compact Sets
- 5.6 Compactness in Infinite Dimensions
- Solutions to Exercises
- References