1 Topics in Functional Analysis -- 1.0 Introduction -- 1.1 Set Theory -- 1.2 Functions -- 1.3 Matrices -- 1.4 Solving Matrix Systems -- 1.5 Metric Spaces -- 1.6 Linear Spaces -- 1.7 Normed Linear Spaces -- 1.8 Approximations -- 2 Integration Theory -- 2.0 Introduction -- 2.1 Reimann and Lebesgue Integrals: Step and Simple Functions -- 2.2 Lebesgue Measure -- 2.3 Measurable Functions -- 2.4 The Lebesgue Integral -- 2.5 Key Theorems in Integration Theory -- 2.6 Lp Spaces -- 2.7 The Metric Space, Lp -- 2.8 Convergence of Sequences -- 2.9 Capsulation -- 3 Hilbert Space and Generalized Fourier Series -- 3.0 Introduction -- 3.1 Inner Product and Hilbert Space -- 3.2 Best Approximations in an Inner Product Space -- 3.3 Approximations in L2(E) -- 3.4 Vector Representations and Best Approximations -- 3.5 Computer Program -- 4 Linear Operators -- 4.0 Introduction -- 4.1 Linear Operator Theory -- 4.2 Operator Norms -- 4.3 Examples of Linear Operators in Engineering -- 4.4 Superposition -- 5 The Best Approximation Method -- 5.0 Introduction -- 5.1 An Inner Product for the Solution of Linear Operator Equations -- 5.2 Definition of Inner Product and Norm -- 5.3 Generalized Fourier Series -- 5.4 Approximation Error Evaluation -- 5.5 The Weighted Inner Product -- 5.6 Considerations in Choosing Basis Functions -- 6 The Best Approximation Method: Applications -- 6.0 Introduction -- 6.1 Sensitivity of Computational Results to Variation in the Inner Product Weighting Factor -- 6.2 Solving Two-Dimensional Potential Problems -- 6.3 Application to Other Linear Operators -- 6.4 Computer Program: Two-Dimensional Potential Problems Using Real Variable Basis Functions -- 6.5 Application of Computer Program -- 7 Solving Potential Problems using the Best Approximation Method -- 7.0 Introduction -- 7.1 The Complex Variable Boundary Element Method -- 7.2 Mathematical Development -- 7.3 The CVBEM and W? -- 7.4 The Space W?A -- 7.5 Applications -- 7.6 Computer Program: Two-Dimensional Potential Problems using Analytic Basis Functions (CVBEM) -- 7.7 Modelling Groundwater Contaminant Transport -- 7.8 Three Dimensional Potential Problems -- 8 Applications to Linear Operator Equations -- 8.0 Introduction -- 8.1 Data Fit Analysis -- 8.2 Ordinary Differential Equations -- 8.3 Best Approximation of Function -- 8.4 Matrix Systems -- 8.5 Linear Partial Differential Equations -- 8.6 Linear Integral Equations -- References -- Appendix A Derivation of CVBEM Approximation Function -- Appendix B Convergence of CVBEM Approximator -- Appendix C The Approximate Boundary for Error Analysis.