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, characteristic polynomial
  • Adjoint of an operator. Self-adjoint, normal, positive definite and unitary operators
  • Spectral theorem for normal operators
  • Schur triangulization lemma
  • SVD-decomposition and pseudoinverse

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
2019-11-25, Franz Luef