Elements of abstract analysis

1. Sets.- 1.1 Set Theory.- 1.2 Relations and Functions.- 1.3 Ordered Sets.- 1.4 Ordinals.- 1.5 The Axiom of Choice.- 2. Counting.- 2.1 Counting Numbers.- 2.2 Cardinality.- 2.3 Enumeration.- 2.4 Cardinality of Unions and Products.- 3. Algebraic Structure.- 3.1 Elementary Algebraic Structures.- 3.2 Vector Spaces.- 3.3 Algebras.- 3.4 Preservation of Algebraic Structure.- 4. Analytic Structure.- 4.1 Ordered Algebraic Structure.- 4.2 Number Systems.- 4.3 Real and Complex Functions.- 4.4 Inequalities.- 5. Linear Structure.- 5.1 Linear Spaces and Algebras.- 5.2 Linear Shapes.- 5.3 Linear Functionals.- 6. Geometric Structure.- 6.1 Semimetrics and Metrics.- 6.2 Seminorms and Norms.- 6.3 Sesquilinear Forms and Inner Products.- 7. Topological Structure.- 7.1 Topologies.- 7.2 Neighbourhoods.- 7.3 Cardinality and Topology.- 7.4 Separation.- 8. Continuity and Openness.- 8.1 Preservation of Topological Structure.- 8.2 Topologies Denned by Functions.- 8.3 Derived Topological Spaces.- 8.4 Topologies on Linear Spaces.- 9. Connectedness.- 9.1 Connected Spaces.- 9.2 Pathwise Connectedness.- 10. Convergence.- 10.1 Filters.- 10.2 Limits.- 11. Compactness.- 11.1 Compact Topological Spaces.- 11.2 Compact Hausdorff Spaces.- 11.3 Local Compactness.- 12. Completeness.- 12.1 Complete Metric Spaces.- 12.2 Banach Spaces.- 12.3 Hilbert Spaces.- 12.4 Banach Algebras.- Solutions.