# Topics

## 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