Course : Mathematical Economics

We have shown that pointwise convergence is strictly weaker than uniform, and we have provided a theorem that complements pointwise convergence with a joint Lipschitz continuity property for the elements of the sequence, and total boundedness of the common domain in order to obtain uniform convergence.

We have examined the usefulness of the strictness of uniform convergence in a general application that concerns the issue approximation of optimization problems (variational approximation). We have proven that uniform convergence implies convergence of optimal values, and by equipping the underlying domain with a metric, w.r.t. which the limit function has a well-distinguishable optimizer, we have extended the convergence to the optimizers. 

We have proceeded with the brief examination of the notion of  completeness.

Notes for the above can be found here, and here. The whiteboards from analogous Ac. year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, and here.

Exercises:

1. The examination of variational approximation was indicatively performed for problems of maximization. Derive the analogous results for minimization.
2. Try to deduce whether pointwise convergence suffices for the approximation of maxima of strictly concave functions. (not trivial!)