Applications of Functional Analysis in Engineering

Functional analysis owes its OrIgms to the discovery of certain striking analogies between apparently distinct disciplines of mathematics such as analysis, algebra, and geometry. At the turn of the nineteenth century, a number of observations, made sporadically over the preceding years, began to inspire systematic investigations into the common features of these three disciplines, which have developed rather independently of each other for so long. It was found that many concepts of this triad-analysis, algebra, geometry-could be incorporated into a single, but considerably more abstract, new discipline which came to be called functional analysis. In this way, many aspects of analysis and algebra acquired unexpected and pro­ found geometric meaning, while geometric methods inspired new lines of approach in analysis and algebra. A first significant step toward the unification and generalization of algebra, analysis, and geometry was taken by Hilbert in 1906, who studied the collection, later called 1 , composed of infinite sequences x = Xb X 2, ... , 2 X , ... , of numbers satisfying the condition that the sum Ik"= 1 X 2 converges. k k The collection 12 became a prototype of the class of collections known today as Hilbert spaces.

1. Physical Space. Abstract Spaces.- Comment 1.1.- 2. Basic Vector Algebra.- Axioms 2.1–2.3 and Definitions 2.1–2.3.- Axioms 2.4–2.8.- Theorem 2.1 (Parallelogram Law).- Problems.- 3. Inner Product of Vectors. Norm.- Definitions 3.1 and 3.2.- Pythagorean Theorem.- Minkowski Inequality.- Cauchy—Schwarz Inequality.- Problems.- 4. Linear Independence. Vector Components. Space Dimension.- Span. Basis. Space Dimension.- Vector Components.- Problems.- 5. Euclidean Spaces of Many Dimensions.- Definitions 5.1–5.6.- Definitions 5.7–5.9.- Orthogonal Projections.- Cauchy—Schwarz and Minkowski Inequalities.- Gram—Schmidt Orthogonalization Process.- lp–Space.- Problems.- 6. Infinite-Dimensional Euclidean Spaces.- Section 6.1. Convergence of a Sequence of Vectors in ??.- Cauchy Sequence.- Section 6.2. Linear Independence. Span, Basis.- Section 6.3. Linear Manifold.- Subspace.- Distance.- Cauchy—Schwarz Inequality.- Remark 6.1.- Problems.- 7. Abstract Spaces. Hilbert Space.- Linear Vector Space. Axioms.- Inner Product.- Pre-Hilbert Space. Dimension. Completeness. Separability.- Metric Space.- Space Ca?t?b and l1.- Normed Spaces. Banach Spaces.- Fourier Coefficients.- Bessel’s Inequality. Parseval’s Equality.- Section 7.1. Contraction Mapping.- Problems.- 8. Function Space.- Hilbert, Dirichlet, and Minkowski Products.- Positive Semi-Definite Metric.- Semi-Norm.- Clapeyron Theorem.- Rayleigh—Betti Theorem.- Linear Differential Operators. Functionals.- Variational Principles.- Bending of Isotropic Plates.- Torsion of Isotropic Bars.- Section 8.1. Theory of Quantum Mechanics.- Problems.- 9. Some Geometry of Function Space.- Translated Subspaces.- Intrinsic and Extrinsic Vectors.- Hyperplanes.- Convexity.- Perpendicularity. Distance.- Orthogonal Projections.- Orthogonal Complement. Direct Sum.- n-Spheres and Hyperspheres.- Balls.- Problems.- 10. Closeness of Functions. Approximation in the Mean. Fourier Expansions.- Uniform Convergence. Mean Square.- Energy Norm.- Space ?2.- Generalized Fourier Series.- Eigenvalue Problems.- Problems.- 11. Bounds and Inequalities.- Lower and Upper Bounds.- Neumann Problem. Dirichlet Integral.- Dirichlet Problem.- Hypercircle.- Geometrical Illustrations.- Bounds and Approximation in the Mean.- Example 11.1. Torsion of an Anisotropic Bar (Numerical Example).- Example 11.2. Bounds for Deflection of Anisotropic Plates (Numerical Example).- Section 11.1. Bounds for a Solution at a Point.- Section 11.1.1. The L*L Method of Kato—Fujita.- Poisson’s Problem.- Section 11.1.2. The Diaz-Greenberg Method.- Example 11.3. Bending a Circular Plate (Numerical Example).- Section 11.1.3. The Washizu Procedure.- Example 11.4. Circular Plate (Numerical Example).- Problems.- 12. The Method of the Hypercircle.- Elastic State Vector.- Inner Product.- Orthogonal Subspaces.- Uniqueness Theorem.- Vertices.- Hypersphere. Hyperplane. Hypercircle.- Section 12.1. Bounds on an Elastic State.- Fundamental and Auxiliary States.- Example 12.1. Elastic Cylinder in Gravity Field (Numerical Example).- Galerkin Method.- Section 12.2. Bounds for a Solution at a Point.- Green’s Function.- Section 12.3. Hypercircle Method and Function Space Inequalities.- Section 12.4. A Comment.- Problems.- 13. The Method of Orthogonal Projections.- Illustrations. Projection Theorem.- Example 13.1. Arithmetic Progression (Numerical Example).- Example 13.2. A Heated Bar (Numerical Example).- Section 13.1. Theory of Approximations. Chebyshev Norm.- Example 13.3. Linear Approximation (Numerical Example).- Problems.- 14. The Rayleigh—Ritz and Trefftz Methods.- Section 14.1. The Rayleigh—Ritz Method.- Coordinate Functions. Admissibility.- Sequences of Functionals.- Lagrange and Castigliano Principles.- Example 14.1. Bounds for Torsional Rigidity.- Example 14.2. Biharmonic Problem.- Section 14.2. The Trefftz Method.- Dirichlet Problem. More General Problem.- Section 14.3. Remark.- Section 14.4. Improvement of Bounds.- Problems.- 15. Function Space and Variational Methods.- Section 15.1. The Inverse Method.- Symmetry and Nondegeneracy of Forms.- Section 15.2. Orthogonal Subspaces.- Minimum Principles.- Section 15.3. Laws’ Approach.- Reciprocal and Clapeyron Theorems.- Minimum Energy Theorem.- Section 15.4. A Plane Tripod.- Lines of Self-Equilibrated Stress and Equilibrium States.- Minimum Principle.- Maximum Principle.- Problems.- 16. Distributions. Sobolev Spaces.- Section 16.1. Distributions.- Delta Function.- Test Functions.- Functionals.- Distribution.- Differentiation of Distributions.- An Example.- Section 16.2. Sobolev Spaces.- Answers to Problems.- References.