**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

**1. Limits and Continuity of Functions**

Definitions of continuity, uniform continuity, properties of continuous

functions, intermediate value theorem, H¨older and Lipschitz continuity.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

**2. Differentiability**

Definitions of differentiability, differentiation rules, differentiable

functions are continuous, higher order derivatives. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

**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* = *γf,* 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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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

**Banach Space**

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

sequence, completeness. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

XII Contents

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

**6. Integrals and Ordinary Differential Equations**

Primitives, Riemann integral, integration rules, integration by parts,

chain rule, mean value theorem, integral and area, ODEs, theorem of

Picard-Lindel¨of on the local existence and uniqueness of solutions of

ODEs with a Lipschitz condition. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

**10. The Implicit Function Theorem. Applications**

Implicit and inverse function theorems, extrema with constraints,

Lagrange multipliers. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Contents XIII

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

**Chapter IV. The Lebesgue Integral**

**12. Preparations. Semicontinuous Functions**

Theorem of Dini, upper and lower semicontinuous functions, the

characteristic function of a set. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

**14. Lebesgue Integrable Functions and Sets**

Upper and lower integral, Lebesgue integral, approximation of

Lebesgue integrals, integrability of sets. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

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

Null functions, null sets, Cantor set, equivalence classes of integrable

functions, the space *L*1*,* Fubini’s theorem for integrable functions.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

**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.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

XIV Contents

**18. The Transformation Formula**

Transformation of multiple integrals under diffeomorphisms, integrals

in polar coordinates. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

**Chapter V. ***Lp* and Sobolev Spaces

**19. The ***Lp*-Spaces

*Lp*-functions, H¨older’s inequality, Minkowski’s inequality,

completeness of *Lp*-spaces, convolutions with local kernels, Lebesgue

points, approximation of *Lp*-functions by smooth functions through

mollification, test functions, covering theorems, partitions of unity.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

**20. Integration by Parts. Weak Derivatives. Sobolev Spaces**

Weak derivatives defined by an integration by parts formula, Sobolev

functions have weak derivatives in *Lp*-spaces, calculus for Sobolev

functions, Sobolev embedding theorem on the continuity of Sobolev

functions whose weak derivatives are integrable to a sufficiently high

power, Poincar´e inequality, compactness theorem of Rellich-Kondrachov

on the *Lp*-convergence of sequences with bounded Sobolev norm.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

**23. Regularity of Weak Solutions**

Smoothness of weakly harmonic functions and of weak solutions of

general elliptic PDEs, boundary regularity, classical solutions. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Contents XV

**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. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

**25. The Eigenvalue Problem for the Laplace Operator**

Eigenfunctions of the Laplace operator form a complete orthonormal

basis of *L*2 as an application of the Rellich compactness theorem.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

**Index** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365