Topologies on Function Spaces. Applications to Functional Differential Equations and Decision Theory.

In the setting of Functional Differential Equations, the \tau_B - topology, a finer one than the Attouch - Wets, was introduced to obtain existence and continuous dependence results [2, 3, 4]. The \tau_B - topology found many applications also in Mathematical Economics [1] as well as in the convergence of dynamic programming models [7]. This topology is natural in metric spaces as it has many good properties without additional conditions on the domain. For example, the \tau_B - topology is always metrizable; under suitable conditions it is also completely metrizable. The class of spaces where that topology has interesting properties is the class of boundedly Atsuji spaces, where the \tau_B - topology restricted to C(X; Y ) coincides with the topology \tau_ucb of uniform convergence on bounded subsets of X. This coincidence works for the Attouch - Wets topology only for a bounded range space, but in the general case it is false. A result of particular importance, among the applications of the \tau_B - topology in Functional Differential Equations Theory, is a theorem concerning a homeomorphism between the space of partial maps (P; \tau_B) and a quotient space of the product of the space of domains equipped with the Attouch-Wets topology and the space of continuous functions equipped with the topology of uniform convergence on bounded sets. The above homeomorphism allows us to reduce the theory of hereditary ordinary differential equations in P to the classical theory in C(X; IR^m) with obvious remarkable advantages. In locally compact second countable metric spaces the topology \tau_B coincides with the topology introduced by Back [1] to study the existence of the so-called jointly continuous utility functions. Back, using a result of Levin, proved the existence of a continuous map from the space of preorders, endowed with the Fell topology, to the space of utility functions (partial maps) endowed with the \tau_B - topology . Recently the results of Levin and Back have been generalized in the case of submetriz- able k_omega- spaces [5, 6]. An example of submetrizable k_omega-space is the space of tempered distributions, which seems to be of interest in the study of market models in the Decision Theory. References [1] K.Back, Concepts of similarity for utility functions, J. Math. Econ., 15, pp. 129-142, (1986). [2] P.Brandi, R.Ceppitelli, Existence, uniqueness and continuous dependence for hereditary differ- ential equations, J. Diff. Equations, 81, pp. 317-339, (1989). [3] P.Brandi, R.Ceppitelli, A new graph topology. Connections with compact open topology, Appl. Analysis, 53, pp.185-196, (1994). [4] P.Brandi, R.Ceppitelli, L. Holà, Boundedly UC spaces and topologies on function spaces, Set Valued Analysis, 16, pp.357-373, (2008). [5] A.Caterino, R.Ceppitelli, F.Maccarino, Continuous utility functions on submetrizable hemicom- pact k-spaces, Applied General Topology, 10, pp.187-195, (2009). [6] A.Caterino, R.Ceppitelli, L. Holà, A generalization of Back's Theorem, preprint. [7] K. J. Langen, Convergence of dynamic programming models, Mathematics of Operations re- search, 6, pp. 493-512, (1981).