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Bounded topologies on Banach spaces and some of their uses in economic theory: a review

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  • Andrew J. Wrobel

Abstract

Known results are reviewed about the bounded and convex bounded variants, bT and cbT, of a topology T on a real Banach space. The focus is on the cases of T = w(P*, P) and of T = m(P*, P), which are the weak* and the Mackey topologies on a dual Banach space P*. The convex bounded Mackey topology, cbm(P*, P), is known to be identical to m(P*, P). As for bm(P*, P), it is conjectured to be strictly stronger than m(P*, P) or, equivalently, not to be a vector topology (except when P is reflexive). Some uses of the bounded Mackey and the bounded weak* topologies in economic theory and its applications are pointed to. Also reviewed are the bounded weak and the compact weak topologies, bw(Y, Y*) and kw(Y, Y*), on a general Banach space Y, as well as their convex variants (cbw and ckw).

Suggested Citation

  • Andrew J. Wrobel, 2020. "Bounded topologies on Banach spaces and some of their uses in economic theory: a review," Papers 2005.05202, arXiv.org, revised Sep 2020.
    • Handle: RePEc:arx:papers:2005.05202
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    1. Horsley, Anthony & Wrobel, Andrzej J, 1991. "The Closedness of the Free-Disposal Hull of a Production Set," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(4), pages 386-391, October.
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