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

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

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

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