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About this book :-
Mathematical Analysis Volume I written by Elias Zakon .
This text is an outgrowth of lectures given at the University of Windsor, Canada. This is updating the undergraduate analysis as a rigorous postcalculus course. While such excellent books as Dieudonn ́e’s Foundations of Modern Analysis are addressed mainly to graduate students, it simplify the modern Bourbaki approach to make it accessible to sufficiently advanced undergraduates.
Zakon developed three volumes on mathematical analysis, which were bound and distributed to students. His goal was to introduce rigorous material as early as possible; later courses could then rely on this material.
Book Detail :-
Title: Mathematical Analysis Volume I
Author(s): Elias Zakon
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About Author :-
The author Elias Zakon was (born 1908) is a Russian Mathematian. Zakon studied mathematics and law in Germany and Poland, and later he joined his father’s law practice in Poland.
In 1956, Zakon moved to Canada and joined as a research fellow at the University of Toronto, he worked with Abraham Robinson. In 1957, he joined the mathematics faculty at the University of Windsor, where the first degrees in the newly established Honours program in Mathematics were awarded in 1960.
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Book Contents :-
Mathematical Analysis Volume I written by Elias Zakon cover the following topics. '
1. Set Theory
Sets and Operations on Sets. Quantifiers, Problems in Set Theory, Relations. Mappings, Problems on Relations and Mappings, Sequences, Some Theorems on Countable Sets, Problems on Countable and Uncountable Sets
2. Real Numbers. Fields
Axioms and Basic Definitions, Natural Numbers. Induction, Problems on Natural Numbers and Induction, Integers and Rationals, Upper and Lower Bounds. Completeness, Problems on Upper and Lower Bounds, Some Consequences of the Completeness Axiom, Powers With Arbitrary Real Exponents. Irrationals, Problems on Roots, Powers, and Irrationals, The Infinities. Upper and Lower Limits of Sequences, Problems on Upper and Lower Limits of Sequences in E*
3. Vector Spaces. Metric Spaces
The Euclidean n-space, En, Problems on Vectors in En, Lines and Planes in En, Problems on Lines and Planes in En, Intervals in En, Problems on Intervals in En, Complex Numbers, Problems on Complex Numbers, Vector Spaces. The Space Cn, Euclidean Spaces, Problems on Linear Spaces, Normed Linear Spaces, Problems on Normed Linear Spaces, Metric Spaces, Problems on Metric Spaces, Open and Closed Sets. Neighborhoods, Problems on Neighborhoods, Open and Closed Sets, Bounded Sets. Diameters, Problems on Boundedness and Diameters, Cluster Points. Convergent Sequences, Problems on Cluster Points and Convergence, Operations on Convergent Sequences, Problems on Limits of Sequences, More on Cluster Points and Closed Sets. Density, Problems on Cluster Points, Closed Sets, and Density, Cauchy Sequences. Completeness, Problems on Cauchy Sequences
4. Function Limits and Continuity
Basic Definitions, Problems on Limits and Continuity, Some General Theorems on Limits and Continuity, More Problems on Limits and Continuity, Operations on Limits. Rational Functions, Problems on Continuity of Vector-Valued Functions, Infinite Limits. Operations in E, Problems on Limits and Operations in E*, Monotone Functions, Problems on Monotone Functions, Compact Sets, Problems on Compact Sets, More on Compactness, Continuity on Compact Sets. Uniform Continuity, Problems on Uniform Continuity; Continuity on Compact Sets, The In, Arcs and Curves. Connected Sets, Problems on Arcs, Curves, and Connected Sets, Product Spaces. Double and Iterated Limits, *Problems on Double Limits and Product Spaces, Sequences and Series of Functions, Problems on Sequences and Series of Functions, Absolutely Convergent Series. Power Series, More Problems on Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 245
5. Differentiation and Antidifferentiation
Derivatives of Functions of One Real Variable, Problems on Derived Functions in One Variable, Derivatives of Extended-Real Functions, Problems on Derivatives of Extended-Real Functions, L’Hˆopital’s Rule, Problems on L’Hˆopital’s Rule, Complex and Vector-Valued Functions on E1, Problems on Complex and Vector-Valued Functions on E1, Antiderivatives (Primitives, Integrals), Problems on Antiderivatives, Differentials. Taylor’s Theorem and Taylor’s Series, Problems on Taylor’s Theorem, The Total Variation (Length) of a Function f : E1 ? E, Problems on Total Variation and Graph Length, Rectifiable Arcs. Absolute Continuity, Problems on Absolute Continuity and Rectifiable Arcs, Convergence Theorems in Differentiation and Integration, Problems on Convergence in Differentiation and Integration, Sufficient Condition of Integrability. Regulated Functions, Problems on Regulated Functions, Integral Definitions of Some Functions, Problems on Exponential and Trigonometric Functions
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