Mid-term, you are expected to know:
- Important theorems, precise assumptions and conclusions;
- The Banach Contraction mapping fixed point theorem;
- Weierstrass theorem;
- Arzela-Ascoli theorem;
- Approximation theorem in finite dimensional space for compact operators;
- The Brouwer fixed point theorem;
- The Schauder fixed point theorem;
- The Leray-Schauder fixed point theorem;
- Sub- and Supersolution's iteration theorem;
- Definitions and concepts;
- Basic definition of a Banach space;
- open set, closed set, compact set, bounded set, convex set, span, convex hall, interior of a set, closure of a set;
- Cauchy sequence, norm, normed space, Banach space;
- Operators, continuous operator, compact operator;
- Problem solving of homework type involving concepts and application of theorems
- Problems involving basic concepts;
- Problems involving application of theorems;
- Integral equations, ODEs, integro-differential equations;
- PDEs (PDE theory will be given to you).
- Other type of homework problem types concerning techniques and concepts.