Functional analysis calculus of variations and numerical methods for models in physics and engineering The book discusses basic concepts of functional analysis measure and integration theory calculus of variations and duality and its applications to variational problems of non convex nature such as the Ginzburg Landau system in superconductivity shape optimization models dual variational formulations for micro magnetism and others. Numerical Methods for such and similar problems such as models in flight mechanics and the Navier Stokes system in fluid mechanics have been developed through the generalized method of lines including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity. The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas 2.9.4 The Closed Graph theorem 2.10 A note on finite dimensional normed spaces 3. Hilbert Spaces 3.1 Introduction 3.2 The main definitions and results 3.3 Orthonormal basis 3.3.1 The Gram Schmidt orthonormalization 3.4 Projection on a convex set 3.5 The theorems of Stampacchia and Lax Milgram 4. The Hahn Banach Theorems and the Weak Topologies 4.1 Introduction 4.2 The Hahn Banach theorems 4.3 The weak topologies 4.4 The weak star topology 4.5 Weak star compactness 4.6 Separable sets 4.7 Uniformly convex spaces 5. Topics on Linear Operators 5.1 Topologies for bounded operators 5.2 Adjoint operators 5.3 Compact operators 5.4 The square root of a positive operator 6. Spectral Analysis a General Approach in Normed Spaces 6.1 Introduction 6.2 Sesquilinear functionals 6.3 About the spectrum of a linear operator defined on a banach space 6.4 The spectral theorem for bounded self adjoint operators 6.4.1 The spectral theorem 6.5 The spectral decomposition of unitary transformations 6.6 Unbounded operators 6.6.1 Introduction 6.7 Symmetric and self adjoint operators 1.5 The Arzela Ascoli theorem 2. Topological Vector Spaces 2.1 Introduction 2.2 Vector spaces 2.3 Some properties of topological vector spaces 2.3.1 Nets and convergence 2.4 Compactness in topological vector spaces 2.4.1 A note on convexity in topological vector spaces 2.5 Normed and metric spaces 2.6 Linear mappings 2.7 Linearity and continuity 2.8 Continuity of operators in Banach spaces 2.9 Some classical results on Banach spaces 2.9.1 The Baire Category theorem 2.9.2 The Principle of Uniform Boundedness 2.9.3 The Open Mapping theorem Cover Title Page Copyright Page Preface Acknowledgements Table of Contents Section I: Functional Analysis 1. Metric Spaces 1.1 Introduction 1.2 The main definitions 1.2.1 The space l∞ 1.2.2 Discrete metric 1.2.3 The metric space s 1.2.4 The space B(A) 1.2.5 The space lp 1.2.6 Some fundamental definitions 1.2.7 Properties of open and closed sets in a metric space 1.2.8 Compact sets 1.2.9 Separable metric spaces 1.2.10 Complete metric spaces 1.3 Completion of a metric space 1.4 Advanced topics on compactness in metric spaces 6.7.1 The spectral theorem using Cayley transform 7. Basic Results on Measure and Integration 7.1 Basic concepts 7.2 Simple functions 7.3 Measures 7.4 Integration of simple functions 7.5 Signed measures 7.6 The Radon Nikodym theorem 7.7 Outer measure and measurability 7.8 Fubinis theorem 7.8.1 Product measures 8. The Lebesgue Measure in Rn 8.1 Introduction 8.2 Properties of the outer measure 8.3 The Lebesgue measure 8.4 Outer and inner approximations of Lebesgue measurable sets 8.5 Some other properties of measurable sets