e-book Introduction to Functional Analysis: Banach Spaces and Differential Calculus

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Table of contents

In chapter 7 we present some formal results on the distributional theory. In the last section some simple examples illustrate the theoretical approach developed. The main focus of chapter 8 is the establishment of basic concepts on Lebesgue and Sobolev spaces.

Bibliographic Information

The results developed include the classical Sobolev imbedding and trace theorems for a special class of domains. In chapter 9 we present basic concepts on the calculus of variations. The main focus of chapter 10 is the development of basic results on variational convex analysis.


  • Functional analysis.
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We address relaxation and duality for both convex and non-convex models. Chapter 11 develops the basic concepts on optimization for variational problems.

Lecture 1 (Part 1): Introduction/Purpose of the functional analysis

Special emphasis is given to models with equality and inequality constraints and respective Lagrange multipliers. Chapter 12 develops duality for a model in finite elasticity. The dual formulations obtained allow the matrix of stresses to be non positive or non negative definite. This is in some sense, an extension of earlier results which establish the complementary energy as a perfect global optimization duality principle only if the stress tensor is positive definite at the equilibrium point.

The results are based on standard tools of convex analysis and the concept of Legendre Transform. Chapter 13 develops dual variational formulations for the two dimensional equations of the nonlinear elastic Kirchhoff-Love plate model.

Notations and Assumptions

We obtain a convex dual variational formulation which allows non positive definite membrane forces. In the third section, similar to the triality criterion introduced in [36], we obtain sufficient conditions of optimality for the present case. Again the results are based on fundamental tools of Convex Analysis and the Legendre Transform, which can easily be analytically expressed for the model in question. Chapters 14 is concerned with existence theory and the development of dual variational formulations for Ginzburg-Landau type equations.

Generalized Uniqueness Theorem for Ordinary Differential Equations in Banach Spaces

Since the primal formulations are non-convex, we use specific results for distance between two convex functions to obtain the dual approaches. Note that we obtain a convex dual formulation for the simpler real case. For such a formulation optimality conditions are also established. The main focus of chapter 15 is the development of a global existence result for the full complex Ginzburg-Landau system in superconductivity.

In a second step we present a closely related optimal control problem and perform its computation by the generalized method of lines. This chapter was published in an article form by Applied Mathematics and Computation-Elsevier , reference [12].

Vogt: Introduction to Functional Analysis. Auflage, Springer Dobrowolski: Angewandte Funktionalanalysis, Springer Heuser: Funktionalanalysis: Theorie und Anwendung, 4. Auflage, Teubner Werner: Funktionalanalysis, 8. Wloka: Funktionalanalysis und Anwendungen, de Gruyter Department of Mathematics.

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Event Calendar. Klausureinsicht Exams can be viewed on Thursday 2nd May from to in The topics we shall study in this lecture include metric spaces notions of topology, compactness continuous linear operators on Banach spaces uniform boundedness principle and open mapping theorem Hilbert spaces, orthonormal bases, Sobolev spaces Dual spaces, Hahn-Banach and Banach-Alaoglu theorems, weak convergence, reflexivity compact linear operators. Exercise sheets 1st exercise Solutions 2nd exercise Solutions 3rd exercise Solutions 4th exercise Solutions 5th exercise Solutions 6th exercise Solutions 7th exercise Solutions 8th exercise Solutions 9th exercise Solutions 10th exercise Solutions 11th exercise Solutions 12th exercise Solutions 13th exercise Solutions 14th exercise Solutions 15th exercise Solutions Test exam Solutions Exam Solutions Examination Written exam is taking place at 20th March , in Added to Your Shopping Cart.

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Generalized Uniqueness Theorem for Ordinary Differential Equations in Banach Spaces

A powerful introduction to one of the most active areas of theoretical and applied mathematics This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented. While occasionally using the more general topological vector space and locally convex space setting, it emphasizes the development of the reader's mathematical maturity and the ability to both understand and "do" mathematics.

This thoughtful, well-organized synthesis of the work of those mathematicians who created the discipline of functional analysis as we know it today also provides a rich source of research topics and reference material. Peter, Minnesota.