The paper studies pure exchange economies with infinite-dimensional commodity spaces in the setting of Riesz dual systems. Several new concepts of equilibrium are introduced. An allocation (x"1,..., x"m) is said to be (a) an Edgeworth equilibrium whenever it belongs to the core of every n-fold replication of the economy; (b) an approximate quasiequilibrium whenever for every @? > 0 there exists some price p @= 0 with p.@w = 1 (where @w = @S@w"i is the total endowment) and with x @>"ix"i implying p.x @>p.@w"i -@?; and (c) an extended Walrasian equilibrium whenever it is a Walrasian equilibrium with respect to an extended price system (i.e., with respect to a price system whose values are extended real numbers). The major results of the paper are the following. (i) If [O, @w] is weakly compact, then Edgeworth equilibria exist. (ii) An allocation is an Edgeworth equilibrium if and only if it is an approximate quasiequilibrium (and also if and only if it is an extended Walrasian equilibrium). (iii) If preferences are uniformly proper, then every Edgeworth equilibrium is a quasiequilibrium. (iv) There exists a two person exchange economy with empty core on C[0, 1] such that preferences are norm continuous, strongly monotone, strictly convex, and uniformly proper, and each agent's endowment is strictly positive.