Preface.

**PART I. ANALYSIS OF FUNCTIONS OF A SINGLE REAL VARIABLE.**

**1. The Real Numbers.**

1.1 Field Axioms.

1.2 Order Axioms.

1.3 Lowest Upper and Greatest Lower Bounds.

1.4 Natural Numbers, Integers and Rational Numbers.

1.5 Recursion, Induction, Summations and Products.

**2. Sequences of Real Numbers.**

2.1 Limits.

2.2 Limit Laws.

2.3 Cauchy Sequences.

2.4 Bounded Sequences.

2.5 Infinite Limits.

**3. Continuous Functions.**

3.1 Limits of Functions.

3.2 Limit Laws.

3.3 One-Sided Limits and Infinite Limits.

3.4 Continuity.

3.5 Properties of Continuous Functions.

3.6 Limits at Infinity.

**4. Differentiable Functions.**

4.1 Differentiability.

4.2 Differentiation Rules.

4.3 Rolle's Theorem and the Mean Value Theorem.

**5. The Riemann Integral I.**

5.1 Riemann Sums and the Integral.

5.2 Uniform Continuity and Integrability of Continuous Functions.

5.3 The Fundamental Theorem of Calculus.

5.4 The Darboux Integral.

**6. Series of Real Numbers I.**

6.1 Series as a Vehicle to Define Infinite Sums.

6.2 Absolute Convergence and Unconditional Convergence.

**7. Some Set Theory.**

7.1 The Algebra of Sets.

7.2 Countable Sets.

7.3 Uncountable Sets.

**8. The Riemann Integral II.**

8.1 Outer Lebesgue Measure.

8.2 Lebesgue's Criterion for Riemann Integrability.

8.3 More Integral Theorems.

8.4 Improper Riemann Integrals.

**9. The Lebesgue Integral.**

9.1 Lebesgue Measurable Sets.

9.2 Lebesgue Measurable Functions.

9.3 Lebesgue Integration.

9.4 Lebesgue Integrals vs. Riemann Integrals.

**10. Series of Real Numbers II.**

10.1 Limits Superior and Inferior.

10.2 The Root Test and the Ratio Test.

10.3 Power Series.

**11. Sequences of Functions.**

11.1 Notions of Convergence.

11.2 Uniform Convergence.

**12. Transcendental Functions.**

12.1 The Exponential Function.

12.2 Sine and Cosine.

12.3 L?H?opital's Rule.

**13. Numerical Methods 203.**

13.1 Approximation with Taylor Polynomials.

13.2 Newton's Method.

13.3 Numerical Integration.

**PART II. ANALYSIS IN ABSTRACT SPACES.**

**14. Integration on Measure Spaces.**

14.1 Measure Spaces.

14.2 Outer Measures.

14.3 Measurable Functions.

14.4 Integration of Measurable Functions.

14.5 Monotone and Dominated Convergence.

14.6 Convergence in Mean, in Measure and Almost Everywhere.

14.7 Product _-Algebras.

14.8 Product Measures and Fubini's Theorem.

**15. The Abstract Venues for Analysis.**

15.1 Abstraction I: Vector Spaces.

15.2 Representation of Elements: Bases and Dimension.

15.3 Identification of Spaces: Isomorphism.

15.4 Abstraction II: Inner Product Spaces.

15.5 Nicer Representations: Orthonormal Sets.

15.6 Abstraction III: Normed Spaces.

15.7 Abstraction IV: Metric Spaces.

15.8 L p Spaces.

15.9 Another Number Field: Complex Numbers.

**16. The Topology of Metric Spaces.**

16.1 Convergence of Sequences.

16.2 Completeness.

16.3 Continuous Functions.

16.4 Open and Closed Sets.

16.5 Compactness.

16.6 The Normed Topology of Rd.

16.7 Dense Subspaces.

16.8 Connectedness.

16.9 Locally Compact Spaces.

**17. Differentiation in Normed Spaces.**

17.1 Continuous Linear Functions.

17.2 Matrix Representation of Linear Functions.

17.3 Differentiability.

17.4 The Mean Value Theorem.

17.5 How Partial Derivatives Fit In.

17.6 Multilinear Functions (Tensors) .

17.7 Higher Derivatives.

17.8 The Implicit Function Theorem.

**18. Measure, Topology and Differentiation.**

18.1 Lebesgue Measurable Sets in Rd.

18.2 C1 and Approximation of Integrable Functions.

18.3 Tensor Algebra and Determinants.

18.4 Multidimensional Substitution.

**19. Manifolds and Integral Theorems.**

19.1 Manifolds.

19.2 Tangent Spaces and Differentiable Functions.

19.3 Differential Forms, Integrals over The Unit Cube.

19.4 k-Forms and Integrals over k-Chains.

19.5 Integration on Manifolds.

19.6 Stokes' Theorem.

**20. Hilbert Spaces.**

20.1 Orthonormal Bases.

20.2 Fourier Series.

20.3 The Riesz Representation Theorem.

**PART III. APPLIED ANALYSIS.**

**21. Physics Background.**

21.1 Harmonic Oscillation.

21.2 Heat and Diffusion.

21.3 Separation of Variables, Fourier Series and Ordinary Differential Equations.

21.4 Maxwell's Equations.

21.5 The Navier Stokes Equation for the Conservation of Mass.

**22. Ordinary Differential Equations.**

22.1 Banach Space Valued Differential Equations.

22.2 An Existence and Uniqueness Theorem.

22.3 Linear Differential Equations.

**23. The Finite Element Method.**

23.1 Ritz-Galerkin Approximation.

23.2 Weakly Differentiable Functions.

23.3 Sobolev Spaces.

23.4 Elliptic Differential Operators.

23.5 Finite Elements.

Conclusion and Outlook.

**APPENDICES.**

**A. Logic.**

A.1 Statements.

A.2 Negations.

**B. Set Theory.**

B.1 The Zermelo-Fraenkel Axioms.

B.2 Relations and Functions.

**C. Natural Numbers, Integers and Rational Numbers.**

C.1 The Natural Numbers.

C.2 The Integers.

C.3 The Rational Numbers.

Bibliography.

Index.