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Chapter 2 Linear Spaces . . . 3 linear manifolds; isomorphic spaces; Cartesian prod ucts equivalence classes;: factor spaces Chapter 3 TopologicalSpaces . 7 convergent sequences: compactness; relative compactness sequen tial copnnes tnuoius functions: inverse mappings; homeomon phisms Chapter 4 Metric Spaces .. . ...... . . . 1 metrics; isometries, Cauchy sequences; completeness; dense subsets; separable metric spaces; completion of a metric space Chapter 5 Normed Linear Spaces ad Banach Spaces 5. norms: bounded subsets Banach spaces; subspaces Chapter 6 ne Prodct Spaces and bert Spaces . . . . ... Inner products; Cauchy-Schwarz inequality; orthogonality EL" and 1 Hilbert spaces: ,i[a,b] and i unit vectors: orthonormal sequences complete orthonormal sequences: separable Hilbert spaces; span of a subset; orthogonal projections; orthogonal complements; orthonormal bases: Pairsevals identity and relation; Fourier coefficients; the Gram- Schmidt process p 7 ,iear unctionals . .. . . ... ..... . 28 unctonals; linear functionals; bounded linear functionals; evaluation funcionals; finie sums deinite integrals; inner products the Riesz epresentation theorerm null spaces; norms; the Hahn-Banach theo- rem; unbounded functionals; conjugate (dual) spaces pr 8 Types of Co ergence in Function Spaces. ..... . 32 srong convergence; weak convergence; pointwise convergence: uni- form convergence; star convergence; weak-star convergence hpter 9 Reproducing Kernel Hilbert Spaces .. ..... .... 35 reproducing kernels; orthogonal project ion; intpolation; approximate integration pter i0 Order Relations in Function Spaces . . . .. .. . 42 reflexiv partial orerings; intervals; interval valued mappings into relexively partially ordered sets lattices; complete lattices; order cnmveence: united extensions subset property of arbitrary map- pinigs the Knaster-Tarski theorem; fixed poins of arbitray map- pings; ine segments in linear spaces; convex sets convex mappings apter Operators in Function Spaces 4 operators; linear operators; nonlinear operators; null spaces: non- sigular linear operators; continuous linear operators bounded ln- ear operators; Neumann series and solution of certain linear opera- tor equations adjoint operators; selfadoint operators; matrix repre- sentations of bounded lnear operators on separable Hilbert spaces; the space L(H,H) of bounded linear operators: types of convergence in LtHI H) Jacobi iteration and Picard iteration linear initl value problems er 12 Comletely Contuous (Compact) Operator... ... 60 completely contuinous operators; Hilbert-Schmidt integral operators: projection operators into finite dimensional subspaces; specal the- oy of completely continuous operators; eigenfunction expansions Gaierkin's method; completely continuous operators in Banach spaces: the Fredholm alternative pr 13 pproximation Methods for Linear Operator Eq ations 68 inite basis methods; finite difference methods: separation of vari- ables and eigenfunction expansons for the diffusion equation rates of convergence; Galerkin's method in Hilbert spaces; collocation methods; finite difference methods: Fredholm integral equations the Nystrom method apter 14 Interal Methods for Operator Equtions . .. . 83 interval arithmetic; interval integration; interval operators; inciu:ion isotonicity; nonlinear operator equations with data perturhations 1apter 15 ontraction Mappings and Iterative Methods 94 ixed point problems; contraction mappings; initial value problems: two-point boundary value problems 1apter 6 Fre het Derivatives . 102 Frchet differentiable operators; locally linear operators the Fr chet derivative; the Gateaux derivative; higher Frechet derivatives the 'aylor theorem in Banach spaces ter 17 Newton's Method in Banach Spaces . .. . ... 116 Newton's iterative method for nonlinear operator equations; loca onvergence; the error squaring property; the Kantorovich theorern computational verification of convergence conditions using inteval analysis; nterval versions of Newton's method apter 8 T ariants of Newton's Method ... . .......... . . . 131 a genera theorem; Ostrowski's theorem; Newton's method; the sim- plified Newton method; the SOR-Newton method (generalied New- ton method); a Gauss-Seidel modification pter 19 Homotopy and Continuation Methods . 1 38 homotopies. successive perturbation methods; continuation methods' curve of zeros: discrete continuation; Davidenkos method compu- tational aspects pter 20 A Hybrid Method for a Free Boundary Problem .......... 146 ts for Selected Exercises .......... . .. ............. . . 160 her R eading ................ . ... . . .......................... 1 73