Springer 1998, 3. Auflage, 2005

Chapter I. Calculus for Functions of One Variable

0. Prerequisites

Properties of the real numbers, limits and convergence of sequences of real numbers, exponential function and logarithm

1. Limits and Continuity of Functions

Definitions of continuity, uniform continuity, properties of continuous functions, intermediate value theorem, Hölder and Lipschitz continuity

2. Differentiability

Definitions of differentiability, differentiation ruies, differentiable functions are continuous, higher order derivatives

3. Characteristic Properties of Differentiable Functions. Differential Equations

Characterization of local extrema by the vanishing of the derivative, mean value theorems, the differential equation f' =  f, uniqueness of solutions of differential equations, qualitative behavior of solutions of differential equations and inequalities, characterization of local maxima and minima via second derivatives, Taylor expansion

4. The Banach Fixed Point Theorem. The Concept of Banach Space

Banach fixed point theorem, definition of norm, metric, Cauchy sequence, completeness

5. Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli

Convergence of sequences of functions, power series, convergence theorems, uniformly convergent sequences, norms on function spaces, theorem of Arzela-Ascoli on the uniform convergence of sequences of uniformly bounded and equicontinuous functions

6. Integrais and Ordinary Differential Equations

Primitives, Riemann integral, integration rules, integration by parts, chain rule, mean value theorem, integral and area, ODEs, theorem of Picard-Lindelöf on the local existence and uniqueness of solutions of ODEs with a Lipschitz condition

Chapter II. Topological Concepts

7. Metric Spaces: Continuity, Topological Notions, Compact Sets

Definition of a metric space, open, closed, convex, connected, compact sets, sequential compactness, continuous mappings between metric spaces, bounded linear operators, equivalence of norms in R^d, definition of a topological space

Chapter III. Calculus in Euclidean and Banach Spaces

8. Differentiation in Banach Spaces

Definition of differentiability of mappings between Banach spaces, differentiation rules, higher derivatives, Taylor expansion

9. Differential Calculus in R^d

A. Scalar valued functions

Gradient, partial derivatives, Hessian, local extrema, Laplace operator, partial differential equations

B. Vector valued functions

Jacobi matrix, vector fields, divergence, rotation

10. The Implicit Function Theorem. Applications

Implicit and inverse function theorems, extrema with constraints, Lagrange multipliers

11. Curves in R^d. Systems of ODEs

Regular and singular curves, length, rectifiability, arcs, Jordan arc theorem, higher order ODE as systems of ODEs

Chapter IV. The Lebesgue Integral

12. Preparations. Semicontinuous Functions.

Theorem of Dini, upper and lower semicontinuous functions, the characteristic function of a set

13. The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets

The integral of continuous and semicontinuous functions, theorem of Fubini, volume, integrals of rotationally symmetric functions and other examples

14. Lebesgue Integrable Functions and Sets

Upper and lower integral, Lebesgue integral, approximation of Lebesgue integrals, integrability of sets

15. Null Functions and Null Sets. The Theorem of Fubini

Null functions, null sets, Cantor set, equivalence classes of integrable functions, the space L¹, Fubini's theorem for integrable functions

16. The Convergence Theorems of Lebesgue Integration Theory

Monotone convergence theorem of B. Levi, Fatou's lemma, dominated convergence theorem of H. Lebesgue, parameter dependent integrals, differentiation under the integral sign

17. Measurable Functions and Sets. Jensen's Inequality. The Theorem of Egorov

Measurable functions and their properties, measurable sets, measurable functions as limits of simple functions, the composition of a measurable function with a continuous function is measurable, Jensen's inequality for convex functions, theorem of Egorov on almost uniform convergence of measurable functions, the abstract concept of a measure.

18. The Transformation Formula

Transformation of multiple integrals under diffeomorphisms, integrals in polar coordinates

Chapter V. L^p and Sobolev Spaces

19. The L^p-Spaces

L^p-functions, Hölder's inequality, Minkowski's inequality, completeness of L^p-spaces, convolutions with local kernels, Lebesgue points, approximation of L^p-functions by smooth functions through mollification, test functions, covering theorems, partitions of unity

20. Integration by Parts. Weak Derivatives. Sobolev Spaces

Weak derivatives defined by an integration by parts formula, Sobolev functions have weak derivatives in L^p-spaces, calculus for Sobolev functions, Sobolev embedding theorem on the continuity of Sobolev functions whose weak derivatives are integrable to a sufficiently high power, Poincaré inequality, compactness theorem of Rellich-Kondrachov on the L^p-convergence of sequences with bounded Sobolev norm

Chapter VI. Introduction to the Calculus of Variations and Elliptic Partial Differential Equations

21. Hilbert Spaces. Weak Convergence

Definition and properties of Hilbert spaces, Riesz representation theorem, weak convergence, weak compactness of bounded sequences, Banach-Saks lemma on the convergence of convex combinations of bounded sequences

22. Variational Principles and Partial Differential Equations

Dirichlet's principle, weakly harmonic functions, Dirichlet problem, Euler-Lagrange equations, variational problems, weak lower semicontinuity of variational integrals with convex integrands, examples from physics and continuum mechanics, Hamilton's principle, equilibrium states, stability, the Laplace operator in polar coordinates

23. Regularity of Weak Solutions

Smoothness of weakly harmonic functions and of weak solutions of general elliptic PDEs, boundary regularity, classical solutions

24. The Maximum Principle

Weak and strong maximum principle for solutions of elliptic PDEs, boundary point lemma of E. Hopf, gradient estimates, theorem of Liouville

25. The Eigenvalue Problem for the Laplace Operator

Eigenfunctions of the Laplace operator form a complete orthonormal basis of L² as an application of the Rellich compactness theorem