Proof of the Second Welfare Theorem, Theorem 5.7*

In this section, I provide a proof of the Second Welfare Theorem. The most important part of the theorem is proved by using the Geometric Hahn-Banach Theorem, Theorem A.28, from Appendix Chapter A

Proof of Theorem 5.7: (Sketch) First, I establish that there exists a price vector p* and an endowment and share allocationthat satisfy conditions 1-3.

(Part 2) I next show that the linear functional φ can be interpreted as a price vector, in particular, that it does have an inner product representation. Consider the functional

where the first line holds by definition, the second line uses the definition of z^φ introduced

above, the third line uses the fact that φ is a linear functional, the fourth line exploits the

so that φ is a continuous linear functional within inner productive presentation.

5.8.