1. Physical Space. Abstract Spaces. - Comment 1. 1. - 2. Basic Vector Algebra. - Axioms 2. 12. 3 and Definitions 2. 12. 3. - Axioms 2. 42. 8. - Theorem 2. 1 (Parallelogram Law). - Problems. - 3. Inner Product of Vectors. Norm. - Definitions 3. 1 and 3. 2. - Pythagorean Theorem. - Minkowski Inequality. - CauchySchwarz Inequality. - Problems. - 4. Linear Independence. Vector Components. Space Dimension. - Span. Basis. Space Dimension. - Vector Components. - Problems. - 5. Euclidean Spaces of Many Dimensions. - Definitions 5. 15. 6. - Definitions 5. 75. 9. - Orthogonal Projections. - CauchySchwarz and Minkowski Inequalities. - GramSchmidt Orthogonalization Process. - lpSpace. - 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. - CauchySchwarz 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. - RayleighBetti 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. - OrthogonalProjections. - 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 KatoFujita. - 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. TheRayleighRitz and Trefftz Methods. - Section 14. 1. The RayleighRitz 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. Language: English