Program (detailed contents):
Sequence of functions
1. Pointwise convergence, uniform convergence
2. Properties of the limit function
3. Approximation: interpolation, density
Applications: interpolation, numerical integration
Series of functions
1. Pointwise, uniform, normal convergence
2. Properties of series of functions
3. Power series
Banach Spaces
1. Cauchy sequences, properties
2. Contraction Mapping theorem
3. Series in Banach spaces
Ordinary Differential Equations: Linear case
1. Examples, affine ODEs
2. Linear ODEs with constant coefficients. Phase portraits. Stability
3. Gronwall Lemma
Introduction to Optimization
1. Convexity: definition, examples
2. Symmetric, positive definite matrices
3. Minimization (compactness arguments). Optimality conditions of order 1 and 2.
Integration
1. Integrals of functions on infinite interval or unbounded functions
2. Integrals with parameters: convergence, continuity, differentiability
3. Integrals of several variables: Fubini theorem, Change of variable.
