Real analysis

<<Cover & Inter-wiki

The real numbers

• Introduction
• Ordered Sets $\mathbb{N}$ and $\mathbb{Z}$
• Ordered Fields $\mathbb{Q}$
• Axioms of The Real Numbers $ℝ$
• Properties of Real Numbers $ℝ$
• Exercises
  • Hints
  • Answers

Sequences and series

• Sequences
• Construction of the Real Numbers $ℝ$
• Series
• Excercises
  • Hints
  • Answers

Limits and Continuity

• Limits
• Continuity
• Total Variation

Differentiation

• Differentiation

Integration

• Riemann integration
• Riemann-Stieltjes integration

Power Series

• Power Series

Sequences of Functions

• Pointwise Convergence
• Uniform Convergence

Multivariate analysis

• Differentiation in Rⁿ
• Inverse function theorem

Appendices

• Bibliography
• Index
• Symbols used in this book

Other suggested topics for inclusion

Since the goal here is to put calculus on a solid footing, I am going to add background, so that we develop the concept of a number first, and work to functions more slowly and methodically, to include Heine-Borel, Weierstrass, etc.

Things that seem to fit in this context:

• (Basic) functional analysis:
  • Uniform Convergence, function spaces
  • Arzela-Ascoli Theorem
  • Stone-Weierstrass Theorem
  • Riemann-Stieltjes integrals and bounded variation
• Measure theory:
  • Measure theory/Lebesgue integrals
• Generalized function (distributions) theory
• Some basic harmonic analysis (Fourier series and transforms).

However, both functional analysis and measure theory could do with their own Wikibooks

Things that might better be in Set Theory:

• Infinite sets and cardinality

Things that might better be in Topology:

• Introduction to different concepts of space: topological, metric, normed, inner product
• Basic topology: accumulation points, closure, interior, boundary, convergence of sequences, in each case with discussion of the type of space the concept is appropriate to.
• Completeness
• Compactness
• Connectedness
• Continuous maps
• Metric spaces
• Contraction Mapping Principle

Analysis