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Constructive Closed Range and Open Mapping Theorems
 Indag. Math. N.S
, 1998
"... We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructive exp ..."
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We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructive exploration of the theory of operators, in particular operators on a Hilbert space ([4], [5], [6]). We work entirely within Bishop's constructive mathematics, which we regard as mathematics with intuitionistic logic. For discussions of the merits of this approach to mathematics in particular, the multiplicity of its modelssee [3] and [10]. The technical background needed in our paper is found in [1] and [9]. Our main aim is to prove the following result, the constructive Closed Range Theorem for operators on a Hilbert space (cf. [16], pages 99103): Theorem 1 Let H be a Hilbert space, and T a linear operator on H such that T exists and ran(T ) is closed. Then ran(T ) and ker(T ) are bo...
Uniform smoothness entails HahnBanach
 Quaestiones Mathematicae
"... Abstract. We show in set theory ZF (without the Axiom of Choice), that uniformly smooth normed spaces satisfy an effective and geometric form of the HahnBanach property. We also compare in ZF the two notions of Gâteaux differentiability and smoothness of a norm, and we obtain a new equivalent of th ..."
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Abstract. We show in set theory ZF (without the Axiom of Choice), that uniformly smooth normed spaces satisfy an effective and geometric form of the HahnBanach property. We also compare in ZF the two notions of Gâteaux differentiability and smoothness of a norm, and we obtain a new equivalent of the HahnBanach axiom. 1.
THE HAHNBANACH PROPERTY AND THE AXIOM OF CHOICE
"... Abstract. We work in the set theory without the axiom of choice: ZF. Though the HahnBanach theorem cannot be proved in ZF, we prove that every Gâteauxdifferentiable uniformly convex Banach space E satisfies the following continuous HahnBanach property: if p is a continuous sublinear functional on ..."
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Abstract. We work in the set theory without the axiom of choice: ZF. Though the HahnBanach theorem cannot be proved in ZF, we prove that every Gâteauxdifferentiable uniformly convex Banach space E satisfies the following continuous HahnBanach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → R is a linear functional such that f ≤ p F, then there exists a linear functional g: E → R, such that g extends f and g ≤ p. We also prove that the continuous HahnBanach property on a topological vector space E is equivalent to the classical geometrical forms of the HahnBanach theorem on E. We then prove that the axiom of Dependent choices (DC) is equivalent to Ekeland’s variational principle, and that it implies the continuous HahnBanach property on Gâteauxdifferentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous HahnBanach property, they do not satisfy the whole HahnBanach property in (ZF+DC). 1
DSGE AND BEYOND – COMPUTABLE AND CONSTRUCTIVE CHALLENGES ♠
, 2011
"... I am, as always, deeply conscious of my debt to my friend and colleague, Stefano Zambelli, in the preparation of this paper. In particular, his recent work on linking coupled nonlinear dynamics at the microeconomic level with aggregate macrodynamics, has been a source of valuable support in the 'bey ..."
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I am, as always, deeply conscious of my debt to my friend and colleague, Stefano Zambelli, in the preparation of this paper. In particular, his recent work on linking coupled nonlinear dynamics at the microeconomic level with aggregate macrodynamics, has been a source of valuable support in the 'beyond ' part of the title of this paper. He is, however, absolved from all responsibilities for the The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the ArrowDebreu General Equilibrium model (ADGE), via Scarf’s Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). An outline of a similar nature, albeit very brie‡y, of the development and structure of AgentBased Economics (ABE) is also included. It is shown that these transformations of the ADGE model are computably and constructively untenable. Suggestions for going ’beyond DSGE and ABE ’ are, then, outlined on the basis of a framework that is underpinned –from the outset –by computability and constructivity considerations.
1 Effective Quantum Observables
, 1995
"... ABSTRACT: Thought experiments about the physical nature of set theoretical counterexamples to the axiom of choice motivate the investigation of peculiar constructions, e.g. an infinite dimensional Hilbert space with a modular quantum logic. Applying a concept due to BENIOFF, we identify the intrinsi ..."
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ABSTRACT: Thought experiments about the physical nature of set theoretical counterexamples to the axiom of choice motivate the investigation of peculiar constructions, e.g. an infinite dimensional Hilbert space with a modular quantum logic. Applying a concept due to BENIOFF, we identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with a failure of the axiom of choice. Here a self adjoint operator is intrinsically effective, iff the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions. Introduction and Summary 0.1. The problem. The formalism of quantum theory is known to depend in an essential way on infinite structures, whence one might wonder about the effects of infinity. A proper analysis of this problem seems to require the reconstruction of quantum theory within some system of constructive mathematics.
ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS
, 2010
"... Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics an ..."
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Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the BolzanoWeierstrass, HahnBanach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels ” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. Acknowledgements: I thank K. Vela Velupillai most particularly for his efforts to push me to consider these matters in the most serious manner, as well as my late father, J. Barkley Rosser [Sr.] and also his friend, the late Stephen C. Kleene, for their personal remarks on these matters to me over a long period of time. I also wish to thank Eric Bach, Ken Binmore, Herb Gintis, Jerome Keisler, Roger Koppl, David Levy, and Adrian Mathias for useful comments. The usual caveat holds. I also wish to dedicate this to K. Vela Velupillai who inspired it with his insistence that I finally deal with the work and thought of my father, J. Barkley Rosser [Sr.], as well as ShuHeng Chen, who supported him in this insistence. I thank both of them for this.