~~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