Topics in Mathematics for economics 2
Instructor: Antonio Villanacci
The course is an independent study and it has “Mathematics for economics” as an indispensable prerequisites. The content of the exam has to be discussed and agreed upon with the instructors. Possible topics are listed below. General topology: Cardinality of sets; definition of topological space and examples; basis and subbasis; sequences; Continuity and different characterizations; topologies generated by functions; metric spaces; first and second countable spaces; separation axioms; compact spaces and characterization in Euclidean, metric and topological space; product spaces, box and product topology; connected spaces; function spaces; pointwise and uniform convergence; the space of continuous functions; compact open topology.
Measure theory: Lebesgue measure theory in Rⁿ; Lebesgue measurable functions; differentiation and integration; Lebesgue integrals and Lp spaces. Functional analysis: normed spaces; Banach space; separable spaces; quotient spaces; equivalent norms; linear continuous functions; images of complete spaces and isometries; finite dimensional space; dual spaces;basic differential calculus in Banach spaces; basic notions of Calculus of Variations and Optimal Control.
Villanacci, A., (2016). Mathematics for Economics 3, Class notes.