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tma4145:2019h:topics [2019-07-12] (nåværende versjon)
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
  
2019-07-12, Franz Luef