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3) is also valid when G = E ∗ as the following simple result shows. 1. If E is a separable Banach space, E ∗ its topological dual, and w ∗ and m ∗ are the topologies defined above, then the following equality holds ∗ B(E m∗ ∗ ) = B(E w ∗ ). ∗ ) ⊆ B(E ∗ ). If G is a member Proof. 3) implies that for each k ≥ 1, G ∩ k B ∗ ∈ B((k B ∗ )w∗ ) = B((k B ∗ )m ∗ ). As already noted, k B ∗ is a member of B(E m∗ ∗ ). Therefore, the restriction of B(E m ∗ ) to k B ∗ consists of the members of B(E m ∗ ) contained in k B ∗ .

Let P be a space of normalized price vectors. Although it is most common to take P = p ∈ R L | p = 1 , where p = L=1 | p |, we only require inf p∈P p > 0. In fact, since there are only two types of commodities, of which the first one is a good and the second one a bad, in our examples, we will take P = { p ∈ R L | p1 = 1}. For each z ∈ R, denote max {z, 0} by z + . Define ψ : P × R L × X × P → R+ by ψ(Q, w, x, p) = | p·(x−w)|+(sup { p · (x − y) | y ∈ X, y Qx, but not x Qy})+. (1) Thus ψ(Q, w, x, p) measures, in monetary terms, the gap between the given consumption vector x ∈ X and the demand of the consumer with the preference relation Q and the initial endowment vector w under the price vector p ∈ P, where the first term penalizes the violation of the budget-balancing condition and the second term penalizes the violation of the utility maximization condition.

2) 26 C. Castaing et al. 4) Indeed, suppose ω ∈ Fq and x ∈ w−ls X nq . There exists a sequence (xk )k≥1 such that x = w − limk→+∞ xk and xk ∈ X n k q (ω), where (X n k q (ω))k≥1 is a subsequence of (X nq (ω))n≥1 . If x = f q (ω) then x ∈ Z q (ω) by the definition of f q . Otherwise, we cannot have xk = f q (ω) for infinitely many indices k. Therefore, xk = f q (ω) for all k ≥ k0 (for some integer k0 ), which yields ω ∈ An k q and xk ∈ X n k (ω) ∩ q (ω) ∩ q B for all k ≥ k0 and, in turn, x ∈ Z q (ω) as well.