# 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