Abstract Calculus: A Categorical Approach

Description

Abstract Calculus: A Categorical Approach provides an abstract approach to calculus. It is intended for graduate students pursuing PhDs in pure mathematics but junior and senior researchers in basically any field of mathematics and theoretical physics will also be interested. Any calculus text for undergraduate students majoring in engineering, mathematics or physics deals with the classical concepts of limits, continuity, differentiability, optimization, integrability, summability, and approximation. This book covers the exact same topics, but from a categorical perspective, making the classification of topological modules as the main category involved. Features Suitable for PhD candidates and researchers Requires prerequisites in set theory, general topology, and abstract algebra, but is otherwise self-contained Dr. Francisco Javier García-Pacheco is a full professor and Director of the Departmental Section of Mathematics at the College of Engineering of the University of Cádiz, Spain. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Attributions Symbol Description I. Functional Calculus 1. Functions 1.1. Set Theory 1.1.1. Predicate logic 1.1.1.1. Syntax 1.1.1.2. Semantics 1.1.1.3. Pragmatics 1.1.2. Axiomatic systems 1.1.2.1. Axioms and theorems 1.1.2.2. Zermelo-Fraenkel 1.2. Relations 1.2.1. Universal structures 1.2.1.1. Multiary relations 1.2.1.2. Binary internal relations 1.2.2. Maps 1.2.2.1. Families and products 1.2.2.2. Composition 1.2.2.3. Hull operators 1.2.3. Equivalence relations 1.2.3.1. Equivalence classes 1.2.3.2. Partitions 1.2.4. Order relations 1.2.4.1. Preorders and orders 1.2.4.2. Lattices 1.2.4.3. Filters, ideals, and bornologies 1.2.4.4. Nets 1.3. Operations 1.3.1. Universal algebras 1.3.1.1. Multiary operations 1.3.1.2. Universal algebras 1.3.1.3. Operators 1.3.1.4. Multioperators 1.3.1.5. Subalgebras 1.3.1.6. Binary internal operations 1.3.2. Groups 1.3.2.1. Magmas 1.3.2.2. Semigroups 1.3.2.3. Monoids 1.3.2.4. Groups 1.3.2.5. Actions 1.3.3. Rings and fields 1.3.3.1. Rings 1.3.3.2. Fields 1.3.4. Modules and algebras 1.3.4.1. Modules 1.3.4.2. Algebras 1.3.5. Effect algebras and Boolean algebras 1.3.5.1. Effect algebras 1.3.5.2. Boolean algebras 1.3.6. Topologies 1.3.6.1. Topological spaces 1.3.6.2. Subsets of a topological space 1.3.6.3. Bases and subbases 1.3.6.4. Separation properties 1.3.6.5. Order topology 1.3.6.6. Convergence topologies 1.3.7. Uniformities 1.3.7.1. Uniform spaces 1.3.7.2. Uniform topologies 1.3.8. Pseudometrics 1.3.8.1. Pseudometric spaces 1.3.8.2. Boundedness 1.3.9. Seminorms 1.3.9.1. Group seminorms 1.3.9.2. Ring seminorms 1.3.9.3. Absolute semivalues 1.3.9.4. Module seminorms 1.3.9.5. Algebra seminorms 2. Limits 2.1. Limits of filters and functions 2.1.1. Limits of filter bases 2.1.1.1. Limits and agglomerations 2.1.1.2. Ultrafilter limits 2.1.1.3. Limits superior and inferior 2.1.2. Limits of functions 2.1.2.1. Filter notion 2.1.2.2. Topological notion 2.1.2.3. Analytical notion 2.1.2.4. One-sided limits 2.1.2.5. Limits superior and inferior 2.1.2.6. Double limits 2.2. Limits of nets and sequences 2.2.1. Limits of nets 2.2.1.1. Filter notion 2.2.1.2. Topological notion 2.2.1.3. Analytical notion 2.2.1.4. Limits of functions through nets 2.2.1.5. Limits superior and inferior 2.2.1.6. Limits of subnets 2.2.1.7. Double limits 2.2.2. Limits of sequences 2.2.2.1. Equivalence of limits 2.2.2.2. Limits of subsequences 2.2.2.3. Cauchy sequences 2.2.2.4. Completeness 2.2.2.5. Completion of a metric space 3. Continuity 3.1. Types of continuity 3.1.1. Pointwise continuity 3.1.1.1. Continuity in topological spaces 3.1.1.2. Open maps 3.1.1.3. Semicontinuity 3.1.2. Uniform continuity 3.1.2.1. Uniform continuity in uniform spaces 3.1.2.2. Index of uniform continuity 3.1.2.3. Lipschitz functions 3.1.2.4. Nonexpansive and contractive functions 3.1.3. Universal topologies 3.1.3.1. Initial topology 3.1.3.2. Final topology 3.2. Topological operations 3.2.1. Topological universal algebras 3.2.1.1. Continuous multiary operations 3.2.1.2. Continuous operators 3.2.2. Topological groups 3.2.2.1. Magma topologies 3.2.2.2. Semigroup topologies 3.2.2.3. Monoid topologies 3.2.2.4. Group topologies 3.2.2.5. Convergence group topologies 3.2.2.6. Projections 3.2.2.7. Seminormed groups 3.2.3. Topological rings 3.2.3.1. Ring topologies 3.2.3.2. Unit zero-neighborhoods 3.2.3.3. Practical rings 3.2.3.4. Topological zero-divisors 3.2.3.5. Convergence ring topologies 3.2.3.6. Closed unit segments 3.2.3.7. Seminormed rings 3.2.3.8. Absolutely semivalued rings 3.2.4. Topological modules 3.2.4.1. Module topologies 3.2.4.2. Boundedness 3.2.4.3. Convergence linear topologies 3.2.4.4. Function spaces 3.2.4.5. Duality 3.2.4.6. Open property 3.2.4.7. Internal points 3.2.4.8. Balancedness 3.2.4.9. Absorbance 3.2.4.10. Convexity 3.2.4.11. Extremality 3.2.4.12. Topological manifolds 3.2.4.13. Seminormed modules 3.2.5. Topological algebras 3.2.5.1. Algebra topologies 3.2.5.2. Seminormed algebras II. Differential Calculus 4. Differentiability 4.1. Derivations 4.1.1. Leibniz’s derivations 4.1.1.1. Product Rule 4.1.1.2. Constants 4.1.2. Rules of derivations 4.1.2.1. Quotient Rule 4.1.2.2. Chain Rule 4.1.2.3. Operations with derivations 4.1.3. Antiderivatives 4.1.3.1. Indefinite integral 4.1.3.2. Linearity 4.1.3.3. Integration by parts 4.1.3.4. Change of variable 4.2. Derivative 4.2.1. Differentiability 4.2.1.1. Directional derivative 4.2.1.2. Differentiable functions 4.2.1.3. Uniformly differentiable functions 4.2.1.4. Derivative of the inversion 4.2.2. Fundamental theorems 4.2.2.1. Continuity 4.2.2.2. Critical points 4.2.2.3. Rolle’s Theorem 4.2.2.4. Chain Rule 4.3. Differential manifolds 4.3.1. Differential local linearity 4.3.1.1. Differentiable atlases 4.3.1.2. Differentiable maps 4.3.2. Tangent space 4.3.2.1. Tangent vectors 4.3.2.2. Tangent space 5. Optimization 5.1. Multiobjective optimization 5.1.1. Formulation 5.1.1.1. General form 5.1.1.2. General solution 5.1.1.3. Pareto optimality 5.1.2. Reformulation 5.1.2.1. Necessity 5.1.2.2. Simplification 5.2. Convex optimization 5.2.1. Convex functions 5.2.1.1. Epigraph 5.2.1.2. Slices 5.2.2. Fundamental theorems 5.2.2.1. Krein-Milman Theorem 5.2.2.2. Bauer Minimum Principle 5.3. Normed optimization 5.3.1. Operator Theory 5.3.1.1. Normalizing rings 5.3.1.2. Operator norms 5.3.1.3. Banach-Alaoglu Theorem 5.3.1.4. Complementation 5.3.2. Supporting vectors 5.3.2.1. Supporting vectors 5.3.2.2. Smoothness 5.3.3. Isometric representations 5.3.3.1. Isometric representations 5.3.3.2. Smooth representations III. Integral Calculus 6. Summability 6.1. Sequences and series 6.1.1. Summability types 6.1.1.1. Convergence 6.1.1.2. Unconditional convergence 6.1.1.3. Subseries convergence 6.1.1.4. Absolute convergence 6.1.1.5. Generalized series 6.1.2. Biorthogonal systems 6.1.2.1. Markushevich bases 6.1.2.2. Schauder bases 6.2. Convergence and summability methods 6.2.1. Sequence spaces 6.2.1.1. Coordinate maps 6.2.1.2. Some sequence spaces 6.2.2. Methods 6.2.2.1. Convergence through free filters 6.2.2.2. Convergence through operators 6.2.2.3. Multipliers 7. Integrability 7.1. Measures 7.1.1. Measures on effect algebras 7.1.1.1. Classification of measures 7.1.1.2. Variation of a measure 7.1.1.3. Probabilities 7.1.2. Measure spaces 7.1.2.1. Simple functions 7.1.2.2. Measurable spaces 7.1.2.3. Measure spaces 7.2. Integration 7.2.1. The definite integral 7.2.1.1. Integrable domains 7.2.1.2. Integral operators IV. Appendix A. Category Theory A.1. Categories A.1.1. Objects and morphisms A.1.2. Special morphisms A.1.3. Special objects A.1.4. Representations A.1.5. Preadditive categories A.1.6. Functors A.2. Universal properties A.2.1. Free objects A.2.2. Product A.2.3. Coproduct A.2.4. Biproduct A.2.5. Tensor product A.2.5.1. Noncommutative case A.2.5.2. Commutative case A.2.5.3. Commutative multilinear case Bibliography Index