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Series that converge on sets of null density. (English) Zbl 0592.40001

Summary: It is shown that a series of positive terms that converges on all sets of null density should be convergent. Using this result we construct examples of complete topological vector spaces that are proper subspaces of a Banach space, but whose dual spaces coincide with the dual space of the Banach space.

MSC:

40A05 Convergence and divergence of series and sequences
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
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References:

[1] Sterling K. Berberian, Lectures in functional analysis and operator theory, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, No. 15. · Zbl 0296.46002
[2] J. L. Kelley and Isaac Namioka, Linear topological spaces, With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. · Zbl 0115.09902
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