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Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Appendix

Part VII Technical appendixes Appendix A. Functional Analysis This appendix provides an introduction to the analysis of functional equations (functional analysis). It describes the contraction mapping theorem, a workhorse for studying dynamic programs. | Part VII Technical appendixes Appendix A. Functional Analysis This appendix provides an introduction to the analysis of functional equations functional analysis . It describes the contraction mapping theorem a workhorse for studying dynamic programs. A.1. Metric spaces and operators We begin with the definition of a metric space which is a pair of objects a set X and a function d.9 Definition A.1.1. A metric space is a set X and a function d called a metric d X x X R. The metric d x y satisfies the following four properties M1. Positivity d x y 0 for all x y G X. M2. Strict positivity d x y 0 if and only if x y. M3. Symmetry d x y d y x for all x y G X. M4. Triangle inequality d x y d x z d z y for all x y and z G X. We give some examples of the metric spaces with which we will be working Example A.1. lp 0 to . We say that X lp 0 to is the set of all sequences of complex numbers x q for which 52 Z q xt p converges where 1 p to . The function dp x y E. xt - yt p 1 p is a metric. Often we will say that p 2 and will work in 12 0 to . Example A.2. lx 0 to . The set X lx 0 to is the set of bounded sequences xt ZQ of real or complex numbers. The metric is dx x y supt xt - yt . Example A.3. Ip to to is the set of two-sided sequences xt t -ix such that 52 Z-x xt p to where 1 p to . The associated metric is dp x y 02t -x xt- ytlp 1 p. 9 General references on the mathematics described in this appendix are Luen-berger 1969 and Naylor and Sell 1982 . 994 - Metric spaces and operators 995 Example A.4. l f-tx to is the set of bounded sequences xt t _ with metric d x y sup xt - yt . Example A.5. Let X C 0 T be the set of all continuous functions mapping the interval 0 T into R. We consider the metric dp x y x t - y t pdt 1 p where the integration is in the Riemann sense. Example A.6. Let X C 0 T be the set of all continuous functions mapping the interval 0 T into R. We consider the metric d x y sup x t - y t . 0 t T We now have the following important definition Definition A.1.2.