On the set of locally convex topologies compatible with a given topology on a vector space: cardinality aspects. (English. Russian original) Zbl 1364.46007
J. Math. Sci., New York 216, No. 4, 577-579 (2016); translation from Sovrem. Mat. Prilozh. 94 (2014).
Summary: For a topological vector space \((X,\tau)\), we consider the family \(\mathrm{LCT}(X,\tau)\) of all locally convex topologies defined on \(X\), which give rise to the same continuous linear functionals as the original topology \(\tau\). We prove that for an infinite-dimensional reflexive Banach space \((X,\tau)\), the cardinality of \(\mathrm{LCT}(X,\tau)\) is at least \(\mathfrak{c}\).
MSC:
| 46A20 | Duality theory for topological vector spaces |
| 46B10 | Duality and reflexivity in normed linear and Banach spaces |
References:
| [1] | F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Grad. Texts Math., 233, Springer-Verlag, New York (2006). · Zbl 1094.46002 |
| [2] | R. Whitley, “Mathematical notes: Projecting <Emphasis Type=”Italic“>m onto <Emphasis Type=”Italic“>c0,” Am. Math. Mon., 73, No. 3, 285-286 (1966). · Zbl 0143.15301 · doi:10.2307/2315346 |
| [3] | W. Sierpiński, Cardinal and Ordinal Numbers, Monogr. Mat., 34, Państowe Wydawnictwo Naukowe, Warszawa (1965). · Zbl 0131.24801 |
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