What is the title of this book intended to signify, what connotations is the adjective Postmodern meant to carry? A potential reader will surely pose this question. To answer it, I should describe what distinguishes the - proach to analysis presented here from what has by its protagonists been called Modern Analysis . Modern Analysis as represented in the works of the Bourbaki group or in the textbooks by Jean Dieudonn e is characterized by its systematic and axiomatic treatment and by its drive towards a high level of abstraction. Given the tendency of many prior treatises on analysis to degenerate into a collection of rather unconnected tricks to solve special problems, this de? nitely represented a healthy achievement. In any case, for the development of a consistent and powerful mathematical theory, it seems to be necessary to concentrate solely on the internal problems and structures and to neglect the relations to other ? elds of scienti? c, even of mathematical study for a certain while. Almost complete isolation may be required to reach the level of intellectual elegance and perfection that only a good mathem- ical theory can acquire. However, once this level has been reached, it can be useful to open one s eyes again to the inspiration coming from concrete external problems.
Inhaltsverzeichnis
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.
