This textbook represents an elementary introduction into applied functional analysis addressed to researchers and beginning graduate students in mathematics, physics and engineering. This work consists of two volumes.

1 The Hahn-Banach Theorem Optimization Problems.- 1.1 The Hahn-Banach Theorem.- 1.2 Applications to the Separation of Convex Sets.- 1.3 The Dual Space C[a,b]*.- 1.4 Applications to the Moment Problem.- 1.5 Minimum Norm Problems and Duality Theory.- 1.6 Applications to ?ebyšev Approximation.- 1.7 Applications to the Optimal Control of Rockets.- 2 Variational Principles and Weak Convergence.- 2.1 The nth Variation.- 2.2 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations.- 2.3 The Lack of Compactness in Infinite-Dimensional Banach Spaces.- 2.4 Weak Convergence.- 2.5 The Generalized Weierstrass Existence Theorem.- 2.6 Applications to the Calculus of Variations.- 2.7 Applications to Nonlinear Eigenvalue Problems.- 2.8 Reflexive Banach Spaces.- 2.9 Applications to Convex Minimum Problems and Variational Inequalities.- 2.10 Applications to Obstacle Problems in Elasticity.- 2.11 Saddle Points.- 2.12 Applications to Duality Theory.- 2.13 The von Neumann Minimax Theorem on the Existence of Saddle Points.- 2.14 Applications to Game Theory.- 2.15 The Ekeland Principle about Quasi-Minimal Points.- 2.16 Applications to a General Minimum Principle via the Palais-Smale Condition.- 2.17 Applications to the Mountain Pass Theorem.- 2.18 The Galerkin Method and Nonlinear Monotone Operators.- 2.19 Symmetries and Conservation Laws (The Noether Theorem).- 2.20 The Basic Ideas of Gauge Field Theory.- 2.21 Representations of Lie Algebras.- 2.22 Applications to Elementary Particles.- 3 Principles of Linear Functional Analysis.- 3.1 The Baire Theorem.- 3.2 Application to the Existence of Nondifferentiable Continuous Functions.- 3.3 The Uniform Boundedness Theorem.- 3.4 Applications to Cubature Formulas.- 3.5 The Open Mapping Theorem.- 3.6 Product Spaces.- 3.7 The Closed Graph Theorem.- 3.8 Applications to Factor Spaces.- 3.9 Applications to Direct Sums and Projections.- 3.10 Dual Operators.- 3.11 The Exactness of the Duality Functor.- 3.12 Applications to the Closed Range Theorem and to Fredholm Alternatives.- 4 The Implicit Function Theorem.- 4.1 m-Linear Bounded Operators.- 4.2 The Differential of Operators and the Fréchet Derivative.- 4.3 Applications to Analytic Operators.- 4.4 Integration.- 4.5 Applications to the Taylor Theorem.- 4.6 Iterated Derivatives.- 4.7 The Chain Rule.- 4.8 The Implicit Function Theorem.- 4.9 Applications to Differential Equations.- 4.10 Diffeomorphisms and the Local Inverse Mapping Theorem.- 4.11 Equivalent Maps and the Linearization Principle.- 4.12 The Local Normal Form for Nonlinear Double Splitting Maps.- 4.13 The Surjective Implicit Function Theorem.- 4.14 Applications to the Lagrange Multiplier Rule.- 5 Fredholm Operators.- 5.1 Duality for Linear Compact Operators.- 5.2 The Riesz-Schauder Theory on Hilbert Spaces.- 5.3 Applications to Integral Equations.- 5.4 Linear Fredholm Operators.- 5.5 The Riesz-Schauder Theory on Banach Spaces.- 5.6 Applications to the Spectrum of Linear Compact Operators.- 5.7 The Parametrix.- 5.8 Applications to the Perturbation of Fredholm Operators.- 5.9 Applications to the Product Index Theorem.- 5.10 Fredholm Alternatives via Dual Pairs.- 5.11 Applications to Integral Equations and Boundary-Value Problems.- 5.12 Bifurcation Theory.- 5.13 Applications to Nonlinear Integral Equations.- 5.14 Applications to Nonlinear Boundary-Value Problems.- 5.15 Nonlinear Fredholm Operators.- 5.16 Interpolation Inequalities.- 5.17 Applications to the Navier-Stokes Equations.- References.- List of Symbols.- List of Theorems.- List of Most Important Definitions.