Contents
Preface
1 The \(l_p\) Spaces
1.1 Some Basic Inequalities
1.2 Practice Problems
2 Compact Metric Spaces
2.1 Limit of a Sequence in \((X,d)\)
2.2 Bounded Sets
2.3 Limit and Continuity of a Function in \((X,d)\)
2.4 Countable Sets
2.5 The Cantor Set
2.6 Partially Ordered Sets
2.7 Totally Bounded Sets
2.8 Compact Sets in a Metric Space
2.9 Sequential Compactness
2.10 Compactness and Continuity
2.11 Practice Problems
3 Complete Metric Spaces
3.1 Dense and Nowhere Dense Sets
3.2 Separable Metric Spaces
3.3 Closed Subspaces and the Nested Set Theorem
3.4 Completion of Metric Spaces
3.5 Fixed Point Theorems and Some Applications
3.6 Elementary Theory of Complete Metric Spaces
3.7 Practice Problems
4 Normed Linear Spaces
4.1 Banach Spaces
4.2 Bounded Linear Transformations
4.3 Dual Spaces
4.4 Practice Problems
5 Inner Product Spaces
5.1 Norms and Orthogonality
5.2 Orthonormal Sets
5.3 Convex Sets and Nearest Points
5.4 Hilbert Spaces
5.5 The Projection Theorem
5.6 The Riesz Representation Theorem
5.7 Complete Orthonormal Sets
5.8 Practice Problems
Prescribed Readings
Recommended Readings
1 The \(l_p\) Spaces
1.1 Some Basic Inequalities
1.2 Practice Problems
2 Compact Metric Spaces
2.1 Limit of a Sequence in \((X,d)\)
2.2 Bounded Sets
2.3 Limit and Continuity of a Function in \((X,d)\)
2.4 Countable Sets
2.5 The Cantor Set
2.6 Partially Ordered Sets
2.7 Totally Bounded Sets
2.8 Compact Sets in a Metric Space
2.9 Sequential Compactness
2.10 Compactness and Continuity
2.11 Practice Problems
3 Complete Metric Spaces
3.1 Dense and Nowhere Dense Sets
3.2 Separable Metric Spaces
3.3 Closed Subspaces and the Nested Set Theorem
3.4 Completion of Metric Spaces
3.5 Fixed Point Theorems and Some Applications
3.6 Elementary Theory of Complete Metric Spaces
3.7 Practice Problems
4 Normed Linear Spaces
4.1 Banach Spaces
4.2 Bounded Linear Transformations
4.3 Dual Spaces
4.4 Practice Problems
5 Inner Product Spaces
5.1 Norms and Orthogonality
5.2 Orthonormal Sets
5.3 Convex Sets and Nearest Points
5.4 Hilbert Spaces
5.5 The Projection Theorem
5.6 The Riesz Representation Theorem
5.7 Complete Orthonormal Sets
5.8 Practice Problems
Prescribed Readings
Recommended Readings