UW-Madison CS 717 - Numerical Functional Analysis (8 pages)

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Numerical Functional Analysis



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Numerical Functional Analysis

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Pages:
8
School:
University of Wisconsin, Madison
Course:
Cs 717 - Numerical Functional Analysis

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Numerical Functional Analysis Carl de Boor draft 30jul03 c 2002 Carl de Boor c 2002 Carl de Boor iii TABLE OF CONTENTS Introduction notation I Preliminaries Linear Algebra linear space linear map special case column maps especially matrices lss s often come as ker or ran quotient space linear functionals dual bidual numerical representation basis Column maps IFn into X use of a basis construction of a basis dimension basic wisdom Row maps X into IFm The interplay between column maps and row maps the inverse of a basis linear projectors factorization and rank The dual of a linear map orthogonality the duals of row maps and column maps use of minimal factorization tests for linear independence Application approximate evaluation of linear functionals interpolation inadequacy of rules rule construction interpolation numerics II Preliminaries Advanced Calculus Topology continuity working with a collection of subsets topology defined equivalent topologies open and closed sets Metric space metric modulus of continuity convergence of sequences Application Contraction maps and fixed point iteration completeness 30jul03 2 5 6 7 8 9 10 11 11 12 13 13 15 16 16 17 18 19 20 20 21 21 22 23 23 24 25 26 29 29 29 29 30 31 33 33 36 37 39 41 c 2003 Carl de Boor iv table of contents summary on contraction Compactness and total boundedness limit points compactness total boundedness examples of compact sets Tykhonov III Normed linear spaces definition norm metric boundedness and continuity bounded below any lm on a finite dimensional domain is continuous closed bounded compact iff finite dimensional computing the norm of a lm Application of map norm Approximate inverse IV The continuous dual hyperplanes and lfl s elimination a useful formula is continuous iff ker is closed error estimates existence of ba from a hyperplane Representation of bounded linear functionals Application Interpolation error and optimal recovery model example interpolation model example cont Lebesgue inequality



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