C.J. Mulvey and J.W. Pelletier. A globalization of the Hahn-Banach theorem. Advances in Mathematics, 89:1-59, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....morphisms : R R preserving positivity and that we have a( a) In presence of classical logic and the axiom of choice, we recover the usual description of Max(R) as a set of points. The important fact is that there are situations where one may fail to have access to the points of Max(R) [BM, MP], for instance without the axiom of choice, or working in intuitionistic logic, while our point free description of Max(R) is still possible . 1.1 Theory of total ordering Let R be an ordered vector space over Q , with a distinguished positive element 1 R . We shall use the letters a; b; c; ....

.... the frame Max(R) generated by the symbols D(a) a 2 R and the relations de ned by Spec r (R) together with the continuity axiom We have D(a) D(b) in Max(R) i for all r 0 there exists s 0 such that D(a r) D(b s) in Spec r (R) The space de ned by Max(R) is compact completely regular [BM, MP] Proof. We have 1 = D(a r) D(s a) if r s and D(a r) D(r a) 0. That Spec r (R) is normal follows then from the corollary 1.7. The proof of this corollary shows that we have r 0 D(a r) in the corresponding compact regular frame, Theorem 1.12 The points of Max(R) can be identi ed with ring ....

Ch. Mulvey and J.W. Pelletier. A globalization of the Hahn-Banach theorem. Adv. Math. 89 (1991), no. 1, 1-59. 16

....function F 7 Gamma F jM is surjective. To state the theorem in a pointfree way, we use the usual denition of surjectivity on points as formal injectivity (see for example [MM92] In this work we were inAEuenced by earlier pointfree proofs of the Hahn Banach theorem by Mulvey and Pelletier in [MP91] and Vermeulen in [Ver86] The proof in [MP91] shows the theorem in any Grothendieck topos and the argument relies on Barr s theorem (which is not justied constructively) and the proof in [Ver86] is done in topos theory with a natural number object (and thus relies on impredicative quantication) ....

.... state the theorem in a pointfree way, we use the usual denition of surjectivity on points as formal injectivity (see for example [MM92] In this work we were inAEuenced by earlier pointfree proofs of the Hahn Banach theorem by Mulvey and Pelletier in [MP91] and Vermeulen in [Ver86] The proof in [MP91] shows the theorem in any Grothendieck topos and the argument relies on Barr s theorem (which is not justied constructively) and the proof in [Ver86] is done in topos theory with a natural number object (and thus relies on impredicative quantication) Our proof, on the other hand, is developed ....

C.J. Mulvey, J.W. Pelletier. A globalization of the Hahn-Banach theorem, Advances in Mathematics 89, pp. 160, 1991.

....natural numbers that can be expressed by an existential formula, and of which the standard proof is based on the existence of a non principal ultrafilter over the natural numbers. This can be compared with the non constructive use of Boolean cover in topos theory in the proof of Barr s theorem [3, 14]. Conclusion We think that the result of theorem 2, that C 0 is an ideal, is interesting in itself. Notice that its proof is quite close to the proof of cut elimination of infinitary propositional logic (see for instance [13] On the other hand, as we already noticed, it can be seen as a ....

Ch. Mulvey and J.W. Pelletier. A Globalization of the Hahn-Banach Theorem. Advances in Mathematics, Vol. 89, (1991), p. 1-59

....of the family of associated spaces. Theorem 29. If D is a normal lattice, the space associated to the normal lattice V (D) is the Vietoris powerlocale of the compact regular space associated to D: The proofs are omitted here. 8 Example: Linear functionals of norm 1 Let E is a seminormed space [MP91] and S be the vector space Q Theta E: Let us write p x an element (p; x) 2 S: Using lemma 2 and theorem 20 we consider the entailment relation over S generated by the axioms Gamma1 x for x 2 N(1) A direct definition is that p i x i q j y j holds iff there exists r i 0; r 0; s j 0 ....

....functionals over E of norm 1: Let E 1 E 2 be two spaces. We have now two entailment relations 1 ; 2 on E 1 ; E 2 respectively. The following result, which is a direct consequence of the direct description of 1 and 2 , can be seen as the localic version of the theorem of Hahn Banach [MP91]. Theorem 32. The entailment relation 2 is a conservative extension of 1 : if x i ; y j 2 E 1 then r i x i 2 s j y j iff r i x i 1 s j y j : Related work and Acknowledgement A Gentzen style sequent calculus is studied in [JKM97] There a category of coherent sequent calculi with ....

C.J. Mulvey and J.W. Pelletier. A globalization of the hahn-banach theorem. Advances in Mathematics, 89:1--59, 1991.

....distributive lattice as the (Tarski )Lindenbaum algebra of a propositional theory. The extension theorem mentionned above becomes then a conservativity theorem between two theories. In this note we analyse the case of the space Fn E: A description can be found in the paper of Mulvey and Pelletier [MP91] or in the work of Vermeulen [Ver86] but we think that our analysis is more perspicuous. In particular, a result of our analysis is a direct and elementary proof of the localic version of the Hahn Banach theorem. The key ingredient is the notion of entailment relation due to D. Scott [Sco74] ....

....C A; x C A C (T ) Theorem 2.2 The relation is the least entailment relation such that x y x; y x; Gammax 3 SemiNormed Space We suppose given a seminormed space E with a seminorm x 2 N(q) for x 2 E and q 2 Q. We shall use freely the concepts and notations presented in the paper [MP91]. We denote by A; B; C; arbitrary finite sets of pairs (x; p) with x 2 E and p 2 Q. We write p x instead of (x; p) We define then the following relation fp 1 x 1 ; p n x n g fq 1 y 1 ; q m y m g which is thought of intuitively as p i u(x i ) q j u(y j ....

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C.J Mulvey and J.W. Pelletier. A globalization of the hahn-banach theorem. Advances in Mathematics, pages 1--59, 1991.

....6. 1 Gamma x) or ( in g i (x) for some n: It can be shown that, conversely, if is a predicate over B that satisfies these properties, then the closed subset that is the intersection of all clopen satisfying is a closed minimal invariant subset. 1. 2 Space of minimal subspace Following [15], we can see the 6 properties as describing forcing conditions on a point of a space. This space M can be seen as an infinitary propositional logic defined inductively by the properties 1. x g(x) 2. 1 1 Gamma x W n ( in g i (x) A point of this space defines then exactly a closed ....

....a predicate over X . Thus, the ideal object that we try to use is a certain predicate over a set of concrete objects, and in most cases, it can be shown that this predicate cannot be defined effectively. A formal space can be described as a set of (forcing) conditions on a point (see for instance [15]) As we have just said, this space may fail to have any effective point. However, even if such points may fail to exist absolutely , they exist always in a relative sense, namely in the sense of the logic defined by the space X: By changing logic , we can then proceed as if a given formal ....

Ch. Mulvey and J.W. Pelletier. (1991) A Globalization of the Hahn-Banach Theorem. Advances in Mathematics, Vol. 89, p. 1-59

....demonstrates that constructive mathematics can be as elegant as classical mathematics. Of more recent work, we could mention the use of point free topology [27, 34, 44] which often makes it possible to replace highly non constructive reasoning involving the axiom of choice by constructive proofs [37, 9]. For a presentation of the fundamental ideas of constructive mathematics we refer to Bishop s book, in particular the first chapter A constructivist manifesto, Dummet [17] Heyting [23] and Troelstra and van Dalen [47, 48] The debate whether mathematics should be built up constructively or ....

Christopher J. Mulvey and Joan Wick Pelletier. A Globalization of the Hahn-Banach Theorem. Advances in Mathematics, 89(1), 1991.

....shows that the norm, in general, is not preserved exactly. In a pointfree formulation of the theorem, one works with finite approximations of the functionals rather than with the functionals themselves. Pointfree formulations of the HahnBanach theorem were presented by Mulvey and Pelletier [13] and by Vermeulen [20] The proof in [13] shows the Hahn Banach theorem in any Grothendieck topos. However, the argument relies on Barr s theorem, which is not justified constructively. The proof in [20] is done in the framework of topos theory with a natural number object, and thus relies on the ....

....is not preserved exactly. In a pointfree formulation of the theorem, one works with finite approximations of the functionals rather than with the functionals themselves. Pointfree formulations of the HahnBanach theorem were presented by Mulvey and Pelletier [13] and by Vermeulen [20] The proof in [13] shows the Hahn Banach theorem in any Grothendieck topos. However, the argument relies on Barr s theorem, which is not justified constructively. The proof in [20] is done in the framework of topos theory with a natural number object, and thus relies on the use of impredicative quantification. The ....

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C.J. Mulvey, J.W. Pelletier. A globalization of the Hahn-Banach theorem, Advances in Mathematics 89, pp. 1-60, 1991.

.... following propositional theory, where the atomic propositions are of the form (f) for each f : N F 2 : 0) 1) f) g) fg) f) 1 Gamma f) g) gf (f) f) Nn [f(n) 1] The geometric theory T ( of a non principal ultrafilter The notion of sheaves over a locale [15] gives a way to make sense of the addition of a generic such object satisfying such forcing conditions T ( Unfortunately, this addition will correspond only to the predicative use of such a : If we see the ultrafilter as defined on decidable subsets of natural numbers, this means that ....

....applies to any Boolean oe algebra. We work then with the locale of oe complete ideals of B: proof is based on the existence of a non principal ultrafilter over the natural numbers. This can be compared with the non constructive use of Boolean cover in topos theory in the proof of Barr s theorem [3, 15]. As we said above, this discussion is left informal, and there is not yet a development of the theory of sheaves over a locale in a setting like constructive type theory. We show however in the next section how to apply concretely these ideas in a given example. 3 A Proof of Ramsey s Theorem As ....

Ch. Mulvey and J.W. Pelletier. A Globalization of the Hahn-Banach Theorem. Advances in Mathematics, Vol. 89, (1991), p. 1-59

....in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly Hahn Banach 1 theorems. Earlier pointfree formulations of the Hahn Banach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986) A constructive proof based on points was given by Bishop (1967) In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As ....

....of this formal topology. Given a subspace M of A, the classical Helly Hahn Banach theorem says that the restriction mapping from the linear functionals on A of norm 1 to those on M is surjective. In terms of covers, conceived as deductive systems, it becomes a conservativity statement (cf. Mulvey and Pelletier 1991): Whenever a is an 1 As explained by Hochstad (1980) the main idea in the usual proof of what is called the Hahn Banach theorem is due to Helly. Since this is also the key idea in our derivation, we here rename the theorem in this way. 2 J. Cederquist, T. Coquand and S. Negri element and U is ....

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Mulvey, C.J. and Pelletier, J.W. (1991). A globalization of the Hahn-Banach theorem, Advances in Mathematics 89, pp. 160.

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C.J. Mulvey and J.W. Pelletier. A globalization of the Hahn-Banach theorem. Advances in Mathematics, 89:1-59, 1991.

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Ch. Mulvey and J.W. Pelletier. A globalization of the Hahn-Banach theorem. Adv. Math. 89 (1991), no. 1, 1--59.

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C.J Mulvey and J.W. Pelletier. A globalization of the hahn-banach theorem. Advances in Mathematics, pages 1--59, 1991.

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