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Separation and weak König’s lemma. (English) Zbl 0922.03079

This article extends the authors’ previous work on the reverse mathematics of Banach space theory, part of which appears in A. J. Humphreys and S. G. Simpson [Trans. Am. Math. Soc. 348, 4231-4255 (1996; Zbl 0857.03036)]. A number of new results concerning separation theorems are presented. In particular, the separation theorem for open convex sets is proved to be equivalent to \(WKL_0\), and the separation theorem for separably closed convex sets is proved to be equivalent to \(ACA_0\). Both logicians and analysts may be interested in the elegant new proof of the Hahn-Banach Theorem in \(WKL_0\). This proof uses techniques similar to those of J. Ł{o}ś and C. Ryll-Nardzewski [Fundam. Math. 38, 233-237 (1951; Zbl 0044.27403)], and is much shorter than the proof previously published by D. K. Brown and S. G. Simpson [Ann. Pure Appl. Logic 31, 123-144 (1986; Zbl 0615.03044)].

MSC:

03F35 Second- and higher-order arithmetic and fragments
46B25 Classical Banach spaces in the general theory
46A55 Convex sets in topological linear spaces; Choquet theory
46B10 Duality and reflexivity in normed linear and Banach spaces
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[1] The Baire category theorem in weak subsytems of second order arithmetic 58 pp 557– (1993)
[2] Introduction to Linear and Convex Programming pp 149– (1985)
[3] Partial realizations of Hilbert’s program 53 pp 349– (1988)
[4] Subsystems of Second Order Arithmetic pp 445– (1998)
[5] Logic and Computation pp 297– (1990)
[6] DOI: 10.1016/0168-0072(85)90030-2 · Zbl 0558.03029 · doi:10.1016/0168-0072(85)90030-2
[7] DOI: 10.1016/0168-0072(90)90068-D · Zbl 0711.03026 · doi:10.1016/0168-0072(90)90068-D
[8] Errett Bishop, Reflections on him and his Research pp 91– (1985)
[9] On Bishop’s Hahn–Banach theorem pp 85– (1985)
[10] Fundamenta Mathematicae 38 pp 233– (1951)
[11] DOI: 10.1090/S0002-9947-96-01725-4 · Zbl 0857.03036 · doi:10.1090/S0002-9947-96-01725-4
[12] Located sets and reverse mathematics pp 37– (1997)
[13] DOI: 10.1016/0168-0072(95)00038-0 · Zbl 0852.03023 · doi:10.1016/0168-0072(95)00038-0
[14] A Course in Functional Analysis pp 399– (1990) · Zbl 0706.46003
[15] DOI: 10.1016/0168-0072(86)90066-7 · Zbl 0615.03044 · doi:10.1016/0168-0072(86)90066-7
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