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Computational functional analysis (Second edition)R E Moore, Ohio State University and M J Cloud, Lawrence Technological University, USA
A stimulating and challenging introduction
(Review of the first edition) SIAM Review, USA, (William W. Hager, Pennsylvania State University).
A very readable introduction, excellent.
The Mathematical Gazette
- contains 100 problem-exercises, answers and tutorial hints for students reading applied functional analysis
- introduces interval analysis into the mainstream of computational functional analysis
This course text fills a gap for first-year graduate-level students reading applied functional analysis or advanced engineering analysis and modern control theory. Containing 100 problem-exercises, answers, and tutorial hints, the first edition is often cited as a standard reference. Making a unique contribution to numerical analysis for operator equations, it introduces interval analysis into the mainstream of computational functional analysis, and discusses the elegant techniques for reproducing Kernel Hilbert spaces. There is discussion of a successful ‘‘hybrid’’ method for difficult real-life problems, with a balance between coverage of linear and non-linear operator equations. The authors successful teaching philosophy: ‘‘We learn by doing’’ is reflected throughout the book.
ISBN 1 904275 24 9
ISBN-13: 978 1 904275 24 4
June 2007
212 pages 234 x 156mm paperback
£50.00 / US$85.00 / €60.00
Usually dispatched within 24 hours
About the authors
Ramon E Moore, Ohio State University, USA. Michael J Cloud, Lawrence Technological University, USA
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Contents
Introduction
Linear spaces
- Linear manifolds
- Isomorphic spaces
- Cartesian products
- Equivalence
- Classes
- Factor spaces
Linear spaces
- Linear manifolds
- Isomorphic spaces
- Cartesian products
- Equivalence
- Classes
- Factor spaces
Topological spaces
- Convergent sequences
- Compactness
- Relative compactness
- Sequen tial compactness
- Ccontinuous functions
- Inverse rnappmgs
- Homeomoo-phisms
Topological spaces
- Convergent sequences
- Compactness
- Relative compactness
- Sequen tial compactness
- Ccontinuous functions
- Inverse rnappmgs
- Homeomoo-phisms
Metric spaces
- Metrics
- Isometries
- Cauchy sequences
- Ccompleteness
- Dense subsets
- Separable metric spaces
- Completion of a metric space
Metric spaces
- Metrics
- Isometries
- Cauchy sequences
- Ccompleteness
- Dense subsets
- Separable metric spaces
- Completion of a metric space
Normed linear spaces and Banach spaces
- Norms
- Bounded subsets
- Banach spaces
- Suhspaces
Normed linear spaces and Banach spaces
- Norms
- Bounded subsets
- Banach spaces
- Suhspaces
Inner product spaces and Hilbert spaces
- Inner products
- Cauchy--Schwarz inequality
- Orthogonality
- E and /2
- Hithert spaces
- Y and l unit vectors
- Orthonormal sequences
- Complete oTth000rmal sequences
- Separable Hubert spaces span of a subset
- Orthogonal projections
- Orthogonal complements
- Orthonormal bases
- Parseval identity and relation
- Fourier coefficients
- The Gram Schmidt process
Inner product spaces and Hilbert spaces
- Inner products
- Cauchy--Schwarz inequality
- Orthogonality
- E and /2
- Hithert spaces
- Y and l unit vectors
- Orthonormal sequences
- Complete oTth000rmal sequences
- Separable Hubert spaces span of a subset
- Orthogonal projections
- Orthogonal complements
- Orthonormal bases
- Parseval identity and relation
- Fourier coefficients
- The Gram Schmidt process
Linear functionals
- Functionals
- Linear functionals
- Bounded linear functionals
- Evaluation functionals
- Finite sums
- Definite integrals
- Inner products
- The Riesz representation theorem
- Null spaces
- Norms
- The Flahn theorem
- Unbounded functionals
- Conjugate (dual) spaces
Linear functionals
- Functionals
- Linear functionals
- Bounded linear functionals
- Evaluation functionals
- Finite sums
- Definite integrals
- Inner products
- The Riesz representation theorem
- Null spaces
- Norms
- The Flahn theorem
- Unbounded functionals
- Conjugate (dual) spaces
Types of convergence in function spaces
- Strong convergence
- Weak convergence
- Pointwise convergence
- Uniform convergence
- Star convergence
- Weak-star convergence
Types of convergence in function spaces
- Strong convergence
- Weak convergence
- Pointwise convergence
- Uniform convergence
- Star convergence
- Weak-star convergence
Reproducing Kernel Hilbert spaces
- Reproducing kernels
- Orthogonal projection
- Interpolation
- Approximate integration
Reproducing Kernel Hilbert spaces
- Reproducing kernels
- Orthogonal projection
- Interpolation
- Approximate integration
Order relations in function spaces
- Reflexive partial orderings
- Intervals
- Interval valued mappings into reflexively partially ordered sets lattices
- Complete lattices
- Order convergence
- United extensions
- Subset property of arbitrary map pings
- The Knaster—Tarski theorem
- Fixed points of arbitrary map pings
- Line segments in linear spaces
- Convex sets
- Convex mappings
Order relations in function spaces
- Reflexive partial orderings
- Intervals
- Interval valued mappings into reflexively partially ordered sets lattices
- Complete lattices
- Order convergence
- United extensions
- Subset property of arbitrary map pings
- The Knaster—Tarski theorem
- Fixed points of arbitrary map pings
- Line segments in linear spaces
- Convex sets
- Convex mappings
Operators in function spaces
- Operators
- Linear operators
- Nonlinear operators
- Null spaces
- Non- singular linear operators
- Continuous linear operators
- Bounded linear operators
- Neumann series and solution of certain linear opera tor equations
- Adjoint operators
- Selfadjoint operators
- Matrix representations of hounded linear operators on separable Hubert spaces
- The space L(H. H) of hounded linear operators types of convergence in L(H . H)
- Jacobi iteration and Picard iteration
- Linear initial value problems
Operators in function spaces
- Operators
- Linear operators
- Nonlinear operators
- Null spaces
- Non- singular linear operators
- Continuous linear operators
- Bounded linear operators
- Neumann series and solution of certain linear opera tor equations
- Adjoint operators
- Selfadjoint operators
- Matrix representations of hounded linear operators on separable Hubert spaces
- The space L(H. H) of hounded linear operators types of convergence in L(H . H)
- Jacobi iteration and Picard iteration
- Linear initial value problems
Completely continuous (compact) operators
- Completely continuous operators
- Hilbert—Schmidt integral operators
- Projection operators into finite dimensional suhspaces
- Spectral the ory of completely continuous operators
- Eigentunction expansions, Galerkms method
- Completely continuous operators in Banach spaces
- The Fredholnn alternative
Completely continuous (compact) operators
- Completely continuous operators
- Hilbert—Schmidt integral operators
- Projection operators into finite dimensional suhspaces
- Spectral the ory of completely continuous operators
- Eigentunction expansions, Galerkms method
- Completely continuous operators in Banach spaces
- The Fredholnn alternative
Approximation methods for linear operator equations
- Finite basis methods
- Finite difference methods
- Separation of variables and eigenfunct expansions for the diffusion equation
- Rates of convergence
- Galerkins method in Hilhert spaces
- Collocation methods
- Finite difference methods
- Fredhoim integral equations
- The Nystrbm method
Approximation methods for linear operator equations
- Finite basis methods
- Finite difference methods
- Separation of variables and eigenfunct expansions for the diffusion equation
- Rates of convergence
- Galerkins method in Hilhert spaces
- Collocation methods
- Finite difference methods
- Fredhoim integral equations
- The Nystrbm method
Interval methods for operator equations
- Interval arithmetic
- Interval integration
- Interval operators
- Inclusion isotonicity
- Nonlinear operator equations with data perturbations
Interval methods for operator equations
- Interval arithmetic
- Interval integration
- Interval operators
- Inclusion isotonicity
- Nonlinear operator equations with data perturbations
Contraction mappings and iterative methods
- Fixed point problems
- Contraction mappings
- Initial value problems
- two-point boundary value problems
Contraction mappings and iterative methods
- Fixed point problems
- Contraction mappings
- Initial value problems
- two-point boundary value problems
Frechet derivatives
- Fréchet differentiable operators
- Locally linear operators
- The Frdchet derivative
- The Gdteaux derivative
- Higher Fréchet derivatives
- The Taylor theorem in Banach spaces
Frechet derivatives
- Fréchet differentiable operators
- Locally linear operators
- The Frdchet derivative
- The Gdteaux derivative
- Higher Fréchet derivatives
- The Taylor theorem in Banach spaces
Newton’s method in Banach spaces
- Convergence
- The error squaring property
- The Kantorovich theorem
- Computational verification of convergence conditions using interval analysis
- Interval versions of Newtons method
Newton’s method in Banach spaces
- Convergence
- The error squaring property
- The Kantorovich theorem
- Computational verification of convergence conditions using interval analysis
- Interval versions of Newtons method
Variants of Newton’s method
- A general theorem
- Ostrowski’s theorem
- Newton’s method
- The simplified Newton method
- The SOR--Newton method (generalized New ton method)
- a Gauss—Seidel modification
Variants of Newton’s method
- A general theorem
- Ostrowski’s theorem
- Newton’s method
- The simplified Newton method
- The SOR--Newton method (generalized New ton method)
- a Gauss—Seidel modification
Homotopy and continuation methods
- Homotopies
- Successive perturbation methods
- Continuation methods
- Curve of zeros
- Discrete continuation
- Davidenko’s method
- Computational aspects
Homotopy and continuation methods
- Homotopies
- Successive perturbation methods
- Continuation methods
- Curve of zeros
- Discrete continuation
- Davidenko’s method
- Computational aspects
A hybrid method for a free boundary problem
A hybrid method for a free boundary problem
Hints for selected exercises
Hints for selected exercises
Further reading
Further reading
Index
Index