Minnesota, and the Stanford Institute for Theoretical Economics. We thank the editor, an

Following Chichilnisky and Chichilnisky-Kalman we establish existence and optimality of competitive equilibrium when commodity spaces are infinite dimen-sional Sobolev spaces, including Hilbert spaces such as weighted LZ which have L. as dense subspaces. We allow general consumption sets with or without lower bounds, thus including securities markets with infinitely many assets and unbounded short sales, and economies with production. We give non-arbitrage conditions on endowments and preferences which suffice for the existence of an equilibrium. Prices are in the same space as commodities. Equilibrium allocations are approximated by allocations in other frequently used spaces such as C(R) and L,.. © 1993 Academic Press, Inc. This paper establishes the existence and Pareto efficiency of competitive equilibrium in Arrow-Debreu exchange economies with infinitely many commodities and a finite number of consumers. It has two distinctive features. One concerns the assumptions on consumption sets: these are general convex sets which may include the whole space or be bounded below. ' This feature allows us to prove existence of an equilibrium in markets without bounds on short sales, in both finite and infinite dimen