Postmodern Analysis

Calculus for Functions of One Variable.- Prerequisites.- Limits and Continuity of Functions.- Differentiability.- Characteristic Properties of Differentiable Functions. Differential Equations.- The Banach Fixed Point Theorem. The Concept of Banach Space.- Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli.- Integrals and Ordinary Differential Equations.- Topological Concepts.- Metric Spaces: Continuity, Topological Notions, Compact Sets.- Calculus in Euclidean and Banach Spaces.- Differentiation in Banach Spaces.- Differential Calculus in $$\mathbb{R}$$ d.- The Implicit Function Theorem. Applications.- Curves in $$\mathbb{R}$$ d. Systems of ODEs.- The Lebesgue Integral.- Preparations. Semicontinuous Functions.- The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets.- Lebesgue Integrable Functions and Sets.- Null Functions and Null Sets. The Theorem of Fubini.- The Convergence Theorems of Lebesgue Integration Theory.- Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov.- The Transformation Formula.- and Sobolev Spaces.- The Lp-Spaces.- Integration by Parts. Weak Derivatives. Sobolev Spaces.- to the Calculus of Variations and Elliptic Partial Differential Equations.- Hilbert Spaces. Weak Convergence.- Variational Principles and Partial Differential Equations.- Regularity of Weak Solutions.- The Maximum Principle.- The Eigenvalue Problem for the Laplace Operator.