Forskjeller

Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.

 — tma4145:2019h:topics [2019-07-12] (nåværende versjon)franzl opprettet 2019-07-12 franzl opprettet 2019-07-12 franzl opprettet Linje 1: Linje 1: + ~~NOTOC~~ + ====== Topics ====== + Below is a list of topics for this course. We might not be able to cover all of them, but the list is a good overview of the contents of the course. + ===== Real numbers ====== + + * Basic properties + * Supremum and infimum of sets + * Open sets, closed sets, neighborhoods ​ + * Cauchy sequences, completeness + * Density of the rational numbers + * Bolzano-Weierstrass theorem + ​ + ===== Normed spaces and innerproduct spaces ===== + + * Vector spaces and norms + * $\mathbb{R}^n,​\mathbb{C}^n$;​ sequence spaces $\ell^1$, $\ell^2$, $\ell^\infty$;​ continuous functions on an interval ​ $C([a,​b]$ ​ + * Innerproduct spaces: Cauchy-Schwarz inequality, parallelogram law, Pythagoras theorem, $\ell^2$ ​ + * Bounded linear transformations between normed spaces and operator norm of a linear transformation,​ the space of bounded operators $B(X,Y)$, condition number + * Cauchy sequences, completeness,​ Banach and Hilbert spaces + * Completeness of $\ell^1$, $\ell^2$, $\ell^\infty$,​ $C([a,b]$ with $\|.\|_\infty$,​ $B(X,Y)$ with operator norm + * Equivalence of norms + ​ + ===== Hilbert spaces ===== + + * Best approximation ​ + * Orthogonal projection, orthogonal decomposition + * Fourier coefficients + * Bessels'​s inequality + * Orthonormal bases and Parseval'​s identity + * Riesz theorem on linear functionals ​ + ​ + ​ + ===== Finite-dimensional vectors spaces and linear transformations ===== + + * Basis, dimension, subspaces, ​ + * Space of polynomials of finite degree, different bases + * Linear transformations and matrices, rank of a linear transformation,​ nullity-rank theorem + * Change-of-basis matrix and similar matrices + * Eigenvalues,​ eigenspaces and generalized eigenspaces + * Caley-Hamilton theorem + * Jordan normal form and its application to linear systems of ODE + * Adjoint of an operator. Self-adjoint,​ normal, positive definite and unitary operators + * Spectral theorem for self-adjoint operators ​ + * LU decomposition + * SVD-decomposition and pseudoinverse + * QR-decomposition + * Power series of operators + + ===== Metric spaces ===== + + * Open and closed sets, neighborhoods,​ Cauchy sequences, completeness + * Continuous functions, uniformly continuous and Lipschitz functions + * Banach fixed point theorem + * Applications of Banach fixed point theorem: systems of equations, Newton iteration, Picard-Lindelöf theorem for ODE