| Preface 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 Index |
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