Table of contents for Computational functional analysis / Ramon E. Moore, Michael J. Cloud.


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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



Library of Congress subject headings for this publication: Functional analysis