# Forskjeller

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

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tma4145:2019h:topics [2019-07-12] (nåværende versjon) franzl opprettet |
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+ | ~~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 | ||