EECS 600 Fall 2004 MWF Lecture Topics - Jeff Fessler This list will be updated regularly (online) over the course of the semester. Course Text: Luenberger 1 policies 1.1-4 overview 2.1-3 vector spaces 2 cartesian product subspaces sum of subsets linear combinations span 2.5 linear independence basis 3 dimension 2.4 convex sets convex hull cones 2.6 norms normed space 2.10 lp Lp spaces holder inequalities 4 minkowski inequalities 2.7 topology distance open spheres interior open sets 5 closure points closed sets bounded sets 2.8 sequences subsequences 6 convergence convergence depends on norm! bounded sequences limit points of sets 2.11 completeness / Banach cauchy sequences 7 cauchy need not converge! complete normed space = Banach space completions closed subets of Banach spaces 8 finite-dim. subspaces of normed spaces 2.9 transformations linearity continuity 9 2.13 compactness 10 upper semicontinuous functions extreme values (Weierstrass) properties of compact sets 11 2.15 dense subsets separable normed spaces Luenberger p 2.20 (skip) (preview of the applications) 10.1 Method of Successive Approximations 12 10.2 Contraction mapping theorem variations on CMT diffeq application example deconvolution example 13 Monotone convergence lemmas preparing for proofs set of subsequence limits connected sets Ostrowski convergence theorem strict monotonicity uniform compactness 14 Meyer convergence theorem nonnegative least squares example 3.1 introduction 3.2 inner product (spaces) 15 Cauchy-Schwrz induced norm parallelogram law Hilbert Spaces Minimum norm problems 16 Chebyshev sets / proximinal sets projectors 3.3 orthogonality pythagorean pre-projection theorem 17 projection theorem (Exam 1 here) (fall break and NSS/MIC) 18 (pradhan) polynomial approximation example orthogonal projection 3.4 orthogonal complements 19 (cancelled - mic) 20 direct sum 21 3.5 orthogonal sets 22 Gram-Schmidt procedure approximation 3.6 normal equations gram matrices 3.9 approximation and Fourier series 23 linear varieties 3.10 dual approximation problem 3.11 (skip: control application) 24 3.12 minimum distance to convex set projection onto convex sets (POCS preview) 25 3.7 Fourier series 3.8 complete orthonormal sequences (review of bases) (skipped) 26 6.1 Linear operators range, null space 6.2 bounded / continuous linear operators B(X,Y): space of bounded linear operators operator norm 27 completeness of B(X,Y) composition of operators (exam 2) 28 6.3 inverse operators linearity of inverses 6.4 Banach inverse theorem (w/o proof) Equivalence of spaces Isomorphic spaces, isomorphisms norm preserving operators linear isometries 29 Isometric isomorphisms unitary equivalence an extension theorem (w/o proof) seperable Hilbert spaces are unitarily equiv to l_2 5.1 dual spaces 5.2 normed dual 30 5.3 Examples of normed duals E^n dual is essentially E^n lp dual is essentially lq for 1 <= p < infty 31 duals of Hilbert spaces 6.5 adjoints in Hilbert spaces 32 properties: linear, bounded, inverse 33 self adjoint, positive definite orthogonal projections are self adjoint unitary operators and adjoints convolution example 34 6.6 relations between the 4 spaces 6.9 normal equations in Hilbert spaces 35 6.10 dual problem: minimum norm solutions 6.11 pseudo-inverse operators 36 geometry of pseudo inverse pseudo-inverse examples l_1 Analysis of the DTFT operator 37 & 38 l_2 Analysis of the DTFT operator 39 7.1 Optimization of Functionals 7.3 Gateaux and Frechet Derivatives 40 no class exam3 ------- Topics that I wish we had had time to cover: 5.4 Hahn-Banach theorem (Extension form) 5.5 Riesz-Representation theorem 5.6 Second dual spaces weak convergence 5.10 continuity of P_K Alignment and orthogonal complements 5.7 Minimum norm problem and its applications 5.8-5.9 Hahn-Banach theorem (Geometric form) and its applications 5.11-5.12 7.4 Conditions for an optimum 7.5 Application to the calculus of variations Conjugate Duality (if time permits) Convex Functionals and their properties 7.8-7.9 Conjugate functionals 7.10 Fenchel duality 7.12 Newton's Method 10.3 (requires Frechet derivatives) existence of isomorphisms ?