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Main Question or Discussion Point
- Author: Carothers
- Title: Real Analysis
- Amazon link https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20
- Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous calculus.
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Metric Spaces
[LIST]
[*] Calculus Review
[LIST]
[*] The Real Numbers
[*] Limits and Continuity
[*] Notes and Remarks
[/LIST]
[*] Countable and Uncountable Sets
[LIST]
[*] Equivalence and Cardinality
[*] The Cantor Set
[*] Monotone Functions
[*] Notes and Remarks
[/LIST]
[*] Metrics and Norms
[LIST]
[*] Metric Spaces
[*] Normed Vector Spaces
[*] More Inequalities
[*] Limits in Metric Spaces
[*] Notes and Remarks
[/LIST]
[*] Open Sets and Closed Sets
[LIST]
[*] Open Sets
[*] Closed Sets
[*] The Relative Metric
[*] Notes and Remarks
[/LIST]
[*] Continuity
[LIST]
[*] Continuous Functions
[*] Homeomorphisms
[*] The Space of Continuous Functions
[*] Notes and Remarks
[/LIST]
[*] Connectedness
[LIST]
[*] Connected Sets
[*] Notes and Remarks
[/LIST]
[*] Completeness
[LIST]
[*] Totally Bounded Sets
[*] Complete Metric Spaces
[*] Fixed Points
[*] Completions
[*] Notes and Remarks
[/LIST]
[*] Compactness
[LIST]
[*] Compact Metric Spaces
[*] Uniform Continuity
[*] Equivalent Metrics
[*] Notes and Remarks
[/LIST]
[*] Category
[LIST]
[*] Discontinuous Functions
[*] The Baire Category Theorem
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] Function Spaces
[LIST]
[*] Sequences of Functions
[LIST]
[*] Historical Background
[*] Pointwise and Uniform Convergence
[*] Interchanging Limits
[*] The Space of Bounded Functions
[*] Notes and Remarks
[/LIST]
[*] The Space of Continuous Functions
[LIST]
[*] The Weierstrass Theorem
[*] Trigonometric Polynomials
[*] Infinitely Differentiable Functions
[*] Equicontinuity
[*] Continuity and Category
[*] Notes and Remarks
[/LIST]
[*] The Stone-Weierstrass Theorem
[LIST]
[*] Algebras and Lattices
[*] The Stone-Weierstrass Theorem
[*] Notes and Remarks
[/LIST]
[*] Functions of Bounded Variation
[LIST]
[*] Functions of Bounded Variation
[*] Helly's First Theorem
[*] Notes and Remarks
[/LIST]
[*] The Riemann-Stieltjes Integral
[LIST]
[*] Weights and Measures
[*] The Riemann-Stieltjes Integral
[*] The Space of Integrable Functions
[*] Integrators of Bounded Variation
[*] The Riemann Integral
[*] The Riesz Representation Theorem
[*] Other Definitions, Other Properties
[*] Notes and Remarks
[/LIST]
[*] Fourier Series
[LIST]
[*] Preliminaries
[*] Dirichlet's Formula
[*] Fejer's Theorem
[*] Complex Fourier Series
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] Lebesgue Measure and Integration
[LIST]
[*] Lebesgue Measure
[LIST]
[*] The Problem of Measure
[*] Lebesgue Outer Measure
[*] Riemann Integrability
[*] Measurable Sets
[*] The Structure of Measurable Sets
[*] A Nonmeasurable Set
[*] Other Definitions
[*] Notes and Remarks
[/LIST]
[*] Measurable Functions
[LIST]
[*] Measurable Functions
[*] Extended Real-Valued Functions
[*] Sequences of Measurable Functions
[*] Approximation of Measurable Functions
[*] Notes and Remarks
[/LIST]
[*] The Lebesgue Integral
[LIST]
[*] Simple Functions
[*] Nonnegative Functions
[*] The General Case
[*] Lebesgue's Dominated Convergence Theorem
[*] Approximation of Integrable Functions
[*] Notes and Remarks
[/LIST]
[*] Additional Topics
[LIST]
[*] Convergence in Measure
[*] The [itex]L_p[/itex] Spaces
[*] Approximation of [itex]L_p[/itex] Functions
[*] More on Fourier Series
[*] Notes and Remarks
[/LIST]
[*] Differentiation
[LIST]
[*] Lebesgue's Differentiation Theorem
[*] Absolute Continuity
[*] Notes and Remarks
[/LIST]
[/LIST]
[*] References
[*] Symbol Index
[*] Topic Index
[/LIST]
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