This paper contains the following results for economies with infinite dimensional commodity spaces. (i) We establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Schmeidler-Hildenbrand [Hildenbrand (1974, p. 33)] finite dimensional theorems. (ii) We provide a counterexample which shows that the above result fails in ordered Banach spaces whose positive cone has an empty interior even if preferences are strictly convex, monotone weakly continuous and initial endowments are strictly positive. (iii) Using the assumption of an 'extremely desirable commodity' (which is automatically satisfied whenever preferences are monotone and the positive cone of the commodity space has a non-empty interior), we establish core-Walras equivalence in any arbitrary separable Banach lattice whose positive cone may have an empty (norm) interior.