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.

This is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory. The third edition contains new material on further important tool in analysis, namely cover theorems. Useful references for such results and further properties of various classes of weakly differential functions are added. And finally, misprints and minor inconsistencies have been corrected.