A note on the Mackey dual of

Author:
Dave Wilkins

Journal:
Proc. Amer. Math. Soc. **118** (1993), 89-92

MSC:
Primary 46E25; Secondary 46A20, 46J10, 54C35

DOI:
https://doi.org/10.1090/S0002-9939-1993-1128731-5

MathSciNet review:
1128731

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Abstract: Let be a compact metric space, and let denote the Mackey topology on with respect to the duality. That is, is the topology of uniform convergence on the weakly compact subsets of . Just as for the weak topology, the dual space of is . However, is very different from weak. Indeed, it is obvious that if is a sequence converging to in , then converges to in the weak topology, yet Kirk has shown (Pacific J. Math. **45** (1973), 543-554) that is closed and discrete in the Mackey topology. We obtain a further result along these lines: For each set . Let denote the totality of all subsets of with the property that . Then a closed set is in iff it is uncountable. Alternatively stated, a closed subset of is countable if and only if there is a weakly compact subset of such that for every pair , there is an with .

**[1]**A. Grothendieck,*Sur les applications linéaires faiblement compactes d’espaces du type 𝐶(𝐾)*, Canadian J. Math.**5**(1953), 129–173 (French). MR**0058866****[2]**Edwin Hewitt and Karl Stromberg,*Real and abstract analysis*, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR**0367121****[3]**R. B. Kirk,*A note on the Mackey topology for (𝐶^{𝑏}(𝑋)*,𝐶^{𝑏}(𝑋))*, Pacific J. Math.**45**(1973), 543–554. MR**0358313****[4]**I. Namioka,*Separate continuity and joint continuity*, Pacific J. Math.**51**(1974), 515–531. MR**0370466****[5]**Georg Schlüchtermann and Robert F. Wheeler,*On strongly WCG Banach spaces*, Math. Z.**199**(1988), no. 3, 387–398. MR**961818**, https://doi.org/10.1007/BF01159786**[6]**Georg Schlüchtermann and Robert F. Wheeler,*The Mackey dual of a Banach space*, Note Mat.**11**(1991), 273–287. Dedicated to the memory of Professor Gottfried Köthe. MR**1258553**

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1128731-5

Article copyright:
© Copyright 1993
American Mathematical Society