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Completeness in the Mackey topology. (English. Russian original) Zbl 1343.46001

Funct. Anal. Appl. 49, No. 2, 97-105 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 21-33 (2015).
For a Banach space \( X \) and a weak\(^*\)-dense norm closed subspace \( P \) of its dual \( X' \), the authors denote by \( \mathcal K \) the set of all absolutely convex weak\(^*\)-compact subsets of \( X' \). Then \( S \) is defined as the set of all continuous linear functionals \( x^* \in X' \) such that \( x^* \) is in the weak\(^*\)-closure of \( K \cap P \) for some \( K \in \mathcal K \) satisfying \( K \subset P \oplus \text{span}\{x^*\} \).
A positive and a negative result about completeness of \( X \) in Mackey topologies is shown. The positive result is:
If \( X \) is complete in the Mackey-topology \( \mu(X,P) \) and \( Y \) is the norm closure of \( P \oplus \text{span}(S_0) \) for some subset \( S_0 \) of \( S \), then \( X \) is complete in the Mackey topology \( \mu(X,Y) \).
The negative result is a converse, namely:
If \( x^* \in X' \setminus S \), then \( X \) is not complete in the Mackey topology \( \mu(X, P \oplus \text{span}\{x^*\}) \).
These results extend work of J. Bonet and B. Cascales [Bull. Aust. Math. Soc. 81, No. 3, 409–413 (2010; Zbl 1205.46004)].

MSC:

46A20 Duality theory for topological vector spaces
46B10 Duality and reflexivity in normed linear and Banach spaces

Citations:

Zbl 1205.46004
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References:

[1] J. Bonet and B. Cascales, “Non-complete Mackey topologies on Banach spaces,” Bull. Aust. Math. Soc., 81:3 (2010), 409-413. · Zbl 1205.46004
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