3. Topology and Convex Optimisation

Chapter 3 covers Topology and convex optimization. In the Chap. 2 we introduced the notion of the scalar product of two vectors in ℜ ⁿ . More generally if a scalar product is defined on some space, then this permits the definition of a norm, or length, associated with a vector, and this in turn allows us to define the distance between two vectors. A distance function or metric may be defined on a space, X, even when X admits no norm. More general than the notion of a metric is that of a topology. This notion allows us to define the idea of continuity of a function as well as analogous ideas for a correspondence. We then introduce three powerful theorems, the Brouwer Fixed Point Theorem for a function, Michael’s Selection Theorem, and the Browder Fixed Point Theorem for a correspondence.