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E-raamat: Abstract Calculus: A Categorical Approach

(Universidad de Cadiz, Spain)
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"Abstract Calculus: A Categorical Approach provides an abstract approach to calculus. It is intended for graduate students pursuing a Ph.D. in Pure Mathematics, but may also find an interested audience among junior and senior researchers in basically anyfield of Mathematics and Theoretical Physics. 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, taking the Category of Topological Modules as the main category involved. Features: Suitable for PhD candidates and researchers. Requires prerequisites in settheory, general topology and abstract algebra, but is otherwise self-contained"--

Abstract Calculus: A Categorical Approach provides an abstract approach to calculus. It is intended for graduate students pursuing a Ph.D. in Pure Mathematics, but may also find an interested audience among junior and senior researchers in basically any field of Mathematics and Theoretical Physics. 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, taking the Category 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.



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.

Preface xv
Attributions xvii
Symbol Description xix
I Functional Calculus 1(238)
1 Functions
3(102)
1.1 Set Theory
3(6)
1.1.1 Predicate logic
3(2)
1.1.1.1 Syntax
3(2)
1.1.1.2 Semantics
5(1)
1.1.1.3 Pragmatics
5(1)
1.1.2 Axiomatic systems
5(4)
1.1.2.1 Axioms and theorems
6(1)
1.1.2.2 Zarmelo-Fraenkel
7(2)
1.2 Relations
9(25)
1.2.1 Universal structures
9(3)
1.2.1.1 Multiary relations
10(1)
1.2.1.2 Binary internal relations
11(1)
1.2.2 Maps
12(6)
1.2.2.1 Families and products
12(2)
1.2.2.2 Composition
14(2)
1.2.2.3 Hull operators
16(2)
1.2.3 Equivalence relations
18(1)
1.2.3.1 Equivalence classes
18(1)
1.2.3.2 Partitions
18(1)
1.2.4 Order relations
19(15)
1.2.4.1 Preorders and orders
19(3)
1.2.4.2 Lattices
22(5)
1.2.4.3 Filters, ideals, and bornologies
27(5)
1.2.4.4 Nets
32(2)
1.3 Operations
34(71)
1.3.1 Universal algebras
34(7)
1.3.1.1 Multiary operations
34(1)
1.3.1.2 Universal algebras
35(2)
1.3.1.3 Operators
37(1)
1.3.1.4 Multioperators
38(1)
1.3.1.5 Subalgebras
38(2)
1.3.1.6 Binary internal operations
40(1)
1.3.2 Groups
41(12)
1.3.2.1 Magmas
41(2)
1.3.2.2 Semigroups
43(1)
1.3.2.3 Monoids
44(4)
1.3.2.4 Groups
48(2)
1.3.2.5 Actions
50(3)
1.3.3 Rings and fields
53(8)
1.3.3.1 Rings
53(6)
1.3.3.2 Fields
59(2)
1.3.4 Modules and algebras
61(6)
1.3.4.1 Modules
61(5)
1.3.4.2 Algebras
66(1)
1.3.5 Effect algebras and Boolean algebras
67(9)
1.3.5.1 Effect algebras
68(5)
1.3.5.2 Boolean algebras
73(3)
1.3.6 Topologies
76(12)
1.3.6.1 Topological spaces
76(1)
1.3.6.2 Subsets of a topological space
77(3)
1.3.6.3 Bases and subbases
80(1)
1.3.6.4 Separation properties
81(3)
1.3.6.5 Order topology
84(3)
1.3.6.6 Convergence topologies
87(1)
1.3.7 Uniformities
88(1)
1.3.7.1 Uniform spaces
89(1)
1.3.7.2 Uniform topologies
89(1)
1.3.8 Pseudoinetrics
89(6)
1.3.8.1 Pseudometric spaces
90(3)
1.3.8.2 Boundedness
93(2)
1.3.9 Seminorms
95(10)
1.3.9.1 Group seminorms
95(2)
1.3.9.2 Ring seminorms
97(2)
1.3.9.3 Absolute semivalues
99(1)
1.3.9.4 Module seminorms
99(5)
1.3.9.5 Algebra seminorms
104(1)
2 Limits
105(28)
2.1 Limits of filters and functions
105(13)
2.1.1 Limits of filter bases
105(4)
2.1.1.1 Limits and agglomerations
105(2)
2.1.1.2 Ultrafilter limits
107(1)
2.1.1.3 Limits superior and inferior
108(1)
2.1.2 Limits of functions
109(9)
2.1.2.1 Filter notion
109(2)
2.1.2.2 Topological notion
111(1)
2.1.2.3 Analytical notion
112(1)
2.1.2.4 One-sided limits
112(1)
2.1.2.5 Limits superior and inferior
113(1)
2.1.2.6 Double limits
114(4)
2.2 Limits of nets and sequences
118(15)
2.2.1 Limits of nets
118(8)
2.2.1.1 Filter notion
118(2)
2.2.1.2 Topological notion
120(1)
2.2.1.3 Analytical notion
120(1)
2.2.1.4 Limits of functions through nets
120(1)
2.2.1.5 Limits superior and inferior
121(1)
2.2.1.6 Limits of subnets
122(2)
2.2.1.7 Double limits
124(2)
2.2.2 Limits of sequences
126(7)
2.2.2.1 Equivalence of limits
126(1)
2.2.2.2 Limits of subsequences
127(1)
2.2.2.3 Cauchy sequences
127(2)
2.2.2.4 Completeness
129(1)
2.2.2.5 Completion of a metric space
130(3)
3 Continuity
133(106)
3.1 Types of continuity
133(14)
3.1.1 Pointwise continuity
133(7)
3.1.1.1 Continuity in topological spaces
133(2)
3.1.1.2 Open maps
135(1)
3.1.1.3 Semicontinuity
136(4)
3.1.2 Uniform continuity
140(3)
3.1.2.1 Uniform continuity in uniform spaces
140(1)
3.1.2.2 Index of uniform continuity
140(1)
3.1.2.3 Lipschitz functions
141(1)
3.1.2.4 Nonexpansive and contractive functions
142(1)
3.1.3 Universal topologies
143(4)
3.1.3.1 Initial topology
143(3)
3.1.3.2 Final topology
146(1)
3.2 Topological operations
147(94)
3.2.1 Topological universal algebras
147(1)
3.2.1.1 Continuous multiary operations
147(1)
3.2.1.2 Continuous operators
148(1)
3.2.2 Topological groups
148(11)
3.2.2.1 Magma topologies
148(1)
3.2.2.2 Semigroup topologies
149(1)
3.2.2.3 Monoid topologies
149(2)
3.2.2.4 Group topologies
151(4)
3.2.2.5 Convergence group topologies
155(3)
3.2.2.6 Projections
158(1)
3.2.2.7 Seminormed groups
158(1)
3.2.3 Topological rings
159(18)
3.2.3.1 Ring topologies
159(3)
3.2.3.2 Unit zero-neighborhoods
162(3)
3.2.3.3 Practical rings
165(1)
3.2.3.4 Topological zero-divisors
165(2)
3.2.3.5 Convergence ring topologies
167(1)
3.2.3.6 Closed unit segments
168(5)
3.2.3.7 Seminormed rings
173(3)
3.2.3.8 Absolutely semivalued rings
176(1)
3.2.4 Topological modules
177(55)
3.2.4.1 Module topologies
177(6)
3.2.4.2 Boundedness
183(2)
3.2.4.3 Convergence linear topologies
185(3)
3.2.4.4 Function spaces
188(3)
3.2.4.5 Duality
191(12)
3.2.4.6 Open property
203(1)
3.2.4.7 Internal points
204(1)
3.2.4.8 Balancedness
205(3)
3.2.4.9 Absorbance
208(4)
3.2.4.10 Convexity
212(13)
3.2.4.11 Extremality
225(4)
3.2.4.12 Topological manifolds
229(1)
3.2.4.13 Seminormed modules
230(2)
3.2.5 Topological algebras
232(9)
3.2.5.1 Algebra topologies
232(3)
3.2.5.2 Seminormed algebras
235(4)
II Differential Calculus 239(62)
4 Differentiability
241(26)
4.1 Derivations
241(7)
4.1.1 Leibniz's derivations
241(2)
4.1.1.1 Product Rule
241(1)
4.1.1.2 Constants
242(1)
4.1.2 Rules of derivations
243(3)
4.1.2.1 Quotient Rule
243(1)
4.1.2.2 Chain Rule
244(2)
4.1.2.3 Operations with derivations
246(1)
4.1.3 Antiderivatives
246(2)
4.1.3.1 Indefinite integral
946
4.1.3.2 Linearity
247(1)
4.1.3.3 Integration by parts
247(1)
4.1.3.4 Change of variable
248(1)
4.2 Derivative
248(12)
4.2.1 Differentiability
248(7)
4.2.1.1 Directional derivative
248(3)
4.2.1.2 Differentiable functions
251(1)
4.2.1.3 Uniformly differentiable functions
252(1)
4.2.1.4 Derivative of the inversion
253(2)
4.2.2 Fundamental theorems
255(5)
4.2.2.1 Continuity
255(1)
4.2.2.2 Critical points
256(1)
4.2.2.3 Rolle's Theorem
256(1)
4.2.2.4 Chain Rule
257(3)
4.3 Differential manifolds
260(7)
4.3.1 Differential local linearity
260(3)
4.3.1.1 Differentiable atlases
260(2)
4.3.1.2 Differentiable maps
262(1)
4.3.2 Tangent space
263(4)
4.3.2.1 Tangent vectors
263(1)
4.3.2.2 Tangent space
264(3)
5 Optimization
267(34)
5.1 Multiobjective optimization
267(9)
5.1.1 Formulation
267(6)
5.1.1.1 General form
267(1)
5.1.1.2 General solution
268(1)
5.1.1.3 Pareto optimality
269(4)
5.1.2 Reformulation
273(3)
5.1.2.1 Necessity
273(1)
5.1.2.2 Simplification
274(2)
5.2 Convex optimization
276(6)
5.2.1 Convex functions
276(5)
5.2.1.1 Epigraph
277(3)
5.2.1.2 Slices
280(1)
5.2.2 Fundamental theorems
281(1)
5.2.2.1 Krein-Milman Theorem
281(1)
5.2.2.2 Bauer Minimum Principle
282(1)
5.3 Normed optimization
282(21)
5.3.1 Operator Theory
282(11)
5.3.1.1 Normalizing rings
283(1)
5.3.1.2 Operator norms
284(7)
5.3.1.3 Banach-Alaoglu Theorem
291(1)
5.3.1.4 Complementation
292(1)
5.3.2 Supporting vectors
293(4)
5.3.2.1 Supporting vectors
293(3)
5.3.2.2 Smoothness
296(1)
5.3.3 Isometric representations
297(6)
5.3.3.1 Isometric representations
297(1)
5.3.3.2 Smooth representations
298(3)
III Integral Calculus 301(44)
6 Summability
303(24)
6.1 Sequences and series
303(15)
6.1.1 Summability types
303(10)
6.1.1.1 Convergence
303(3)
6.1.1.2 Unconditional convergence
306(1)
6.1.1.3 Subseries convergence
307(1)
6.1.1.4 Absolute convergence
308(3)
6.1.1.5 Generalized series
311(2)
6.1.2 Biorthogonal systems
313(5)
6.1.2.1 Markushevich bases
313(2)
6.1.2.2 Schauder bases
315(3)
6.2 Convergence and summability methods
318(9)
6.2.1 Sequence spaces
319(1)
6.2.1.1 Coordinate maps
319(1)
6.2.1.2 Some sequence spaces
320(1)
6.2.2 Methods
320(7)
6.2.2.1 Convergence through free filters
321(3)
6.2.2.2 Convergence through operators
324(1)
6.2.2.3 Multipliers
325(2)
7 Integrability
327(18)
7.1 Measures
327(14)
7.1.1 Measures on effect algebras
327(9)
7.1.1.1 Classification of measures
328(3)
7.1.1.2 Variation of a measure
331(4)
7.1.1.3 Probabilities
335(1)
7.1.2 Measure spaces
336(5)
7.1.2.1 Simple functions
336(1)
7.1.2.2 Measurable spaces
337(4)
7.1.2.3 Measure spaces
341(1)
7.2 Integration
341(6)
7.2.1 The definite integral
342(5)
7.2.1.1 Integrable domains
342(1)
7.2.1.2 Integral operators
342(3)
IV Appendix 345(20)
A Category Theory
347(18)
A.1 Categories
347(7)
A.1.1 Objects and morphisms
347(2)
A.1.2 Special morphisms
349(1)
A.1.3 Special objects
350(2)
A.1.4 Representations
352(1)
A.1.5 Preadditive categories
352(1)
A.1.6 Functors
353(1)
A.2 Universal properties
354(11)
A.2.1 Free objects
354(1)
A.2.2 Product
355(1)
A.2.3 Coproduct
356(1)
A.2.4 Biproduct
357(1)
A.2.5 Tensor product
358(7)
A.2.5.1 Noncommutative case
358(2)
A.2.5.2 Commutative case
360(2)
A.2.5.3 Commutative multilinear case
362(3)
Bibliography 365(6)
Index 371
Prof. Dr. Francisco Javier García-Pacheco graduated in Mathematics from the University of Cádiz (Spain, EU) in 2000. He later become an Adjunct Professor at the University of Cádiz and did his doctorate in Pure Mathematics, under the tutelage of Prof. Dr. Antonio Aizpuru, defending his dissertation on May 27, 2005.

One year before defending the thesis, in August 2004, he was awarded a Graduate Teaching Assistantship at Kent State University (Ohio, USA) to do a second doctorate in Nonlinear Analysis, under the tutelage of Prof. Dr. Richard Aron. He defended this second dissertation on January 19, 2007, obtaining the outstanding dissertation award. During the first two years of his stay at Kent State University, he was hired as a Graduate Teaching Assistant. His last year he was awarded with a Teaching Fellowship.

In August 2007, he was hired by Texas A&M University (Texas, USA) as a Visiting Assistant Professor until August 2010, under the tutelage of Prof. Dr. Bill Johnson.

Since September 2010 he has been at the University of Cádiz, first as an Assistant Professor (Sep 2010 - May 2012), then as an Associate Professor (June 2012 - Feb 2019), and finally as a Full Professor (March 2019 - present). During the Summer semester of 2011, he did a research and teaching stay at Texas A& M University-Central Texas as Visiting Associate Professor.

This international experience in research and teaching made possible for him to have authored more than 100 research manuscripts.

He also has experience on academic management. From December 2017 through July 2019, he held the academic position of Director of the Secretariat for International Projects since December 2017 until July 2019. Since March 2020, he is currently Director of the Departmental Section of Mathematics at the College of Engineering of the University of Cádiz.
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