Topology and Convex Optimisation

Norman Schofield
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Norman Schofield: Washington University in Saint Louis

Chapter 3 in Mathematical Methods in Economics and Social Choice, 2014, pp 77-133 from Springer

Abstract: Abstract 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.

Keywords: Nash Equilibrium; Topological Space; Open Cover; Topological Vector Space; Product Topology (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sptchp:978-3-642-39818-6_3

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DOI: 10.1007/978-3-642-39818-6_3

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