Constructive and computable Hahn–Banach theorems for the (second) fundamental theorem of welfare economics☆
Section snippets
A preamble
“There are many–denumerably, I suspect–other versions of the [Hahn–Banach] theorem, …. It has many applications not only outside functional analysis but outside mathematics”.
Narici (2007, p. 88); italics added.
The three ‘crown jewels’ of the mathematical economics of the second half of the twentieth century are undoubtedly the proof of the existence of a Walrasian Exchange Equilibrium and the mathematically rigorous demonstration of the validity of the two fundamental theorems of welfare
Constructive and computable Hahn–Banach Theorems
“Throughout [Computability in Analysis and Physics] we have attempted to give general principles from which the effectivization or noneffectiveness of well-known classical theorems follow as corollaries …. For example, it would be interesting to have a general principle which gave as corollary the known facts concerning the Hahn–Banach Theorem”. Pour-El and Richards (1989, p. 194).
Four constructive and computable versions of the H–B T are now given. The first two, are referred to as Theorem B,
Concluding notes
“…[L]et us return to the issue of computation over the reals as a foundation for scientific computation, aka computational science or numerical analysis. …
[I]n practice, those computations are made11 in ‘floating point arithmetic’ using finite decimals with relatively few significant digits, for which computation
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Cited by (0)
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I am greatly indebted to an anonymous referee and the Editor of this Journal for valuable and encouraging comments on an earlier version of this paper. The rewriting of the original submission, facilitated also by the generous time given by the Editor, owes a great deal to the help and advice of my able graduate student, Ilker Aslantepe (who should, by all standards of ethical behavior, be a joint author). I am solely responsible for all remaining infelicities.