Silvia Evelyn M. Companion Notes A Working Excursion to Accompany Baby Rudin

1999. 434 p.These notes have been prepared to assist students who are learning Advanced Cal -culus/Real Analysis for the ¿ rst time in courses or self-study programs that are using the text Principles of Mathematical Analysis (3rd Edition) by Walter Rudin. References to page numbers or general location of results that mention our text are always referring to Rudin’s book. The notes are designed to encourage or engender an interactive approach to learning the material, provide more examples at the introductory level, offer some alternative views of some of the concepts, and draw a clearer connection to the mathematics that is prerequisite to under-standing the development of the mathematical analysisAbout the Organization of the Material
About the Errors
The Field of Reals and Beyond
Fields
Ordered Fields
Special Subsets of an Ordered Field
Bounding Properties
The Real Field
Density Properties of the Reals
Existence of nth Roots
The Extended Real Number System
The Complex Field
Thinking Complex
Problem Set A
From Finite to Uncountable Sets
Some Review of Functions
A Review of Cardinal Equivalence
Denumerable Sets and Sequences
Review of Indexed Families of Sets
Cardinality of Unions Over Families
The Uncountable Reals
Problem Set B
Metric Spaces and Some Basic Topology
Euclidean n –space Metric Spaces
Point Set Topology on Metric Spaces
Complements and Families of Subsets of Metric Spaces
Open Relative to Subsets of Metric Spaces
Compact Sets
Compactness in Euclidean n -space
Connected Sets
Perfect Sets
Problem Set C
Sequences and Series –First View
Sequences and Subsequences in Metric Spaces
Cauchy Sequencesin Metric Spaces
Sequences in Euclidean k -space
Upper and Lower Bounds
Some Special Sequences
Series of Complex Numbers
Some (Absolute) Convergence Tests
Absolute Convergence and Cauchy Products
Hadamard Products and Series with Positive and Negative Terms
Discussing Convergence
Rearrangements of Series
Problem Set D
Functions on Metric Spaces and Continuity
Limits of Functions
Continuous Functionson Metric Spaces
A Characterization of Continuity
Continuity and Compactness
Continuity and Connectedness
Uniform Continuity
Discontinuities and Monotonic Functions
Limits of Functions in the Extended Real Number System
Problem Set E
Differentiation: Our First View
The Derivative
Formulas for Differentiation
Revisiting A Geometric Interpretation for the Derivative
The Derivative and Function Behavior
Continuity (or Discontinuity) of Derivatives
The Derivative and Finding Limits
Inverse Functions
Derivatives of Higher Order
Differentiation of Vector-Valued Functions
Problem Set F
Riemann-Stieltjes Integration
Riemann Sums and Integrability
Properties of Riemann-Stieltjes Integrals
Riemann Integrals and Differentiation
Some Methods of Integration
The Natural Logarithm Function
ntegration of Vector-Valued Functions
Recti¿ able Curves
Problem Set G
Sequences and Series of Functions
Pointwise and Uniform Convergence
Sequences of Complex-Valued Functions on Metric Spaces
Conditions for Uniform Convergence
Property Transmission and Uniform Convergence
Families of Functions
The Stone-Weierstrass Theorem
Problem Set H
Some Special Functions
Power Series Over the Reals
Some General Convergence Properties
Designer Series
Another Visit With the Logarithm Function
A Series Development of Two Trigonometric Functions
Series from Taylor’s Theorem Some Series To Know & Love
Series From Other Series
Fourier Series
Problem Set I