Mid-term, you are expected to know:

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