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# Postmodern Analysis

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

L^{1}, 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 ofL^{p}-spaces, convolutions with local kernels, Lebesgue points, approximation ofL^{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 theL^{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

L^{2}as an application of the Rellich compactness theorem