TY - JOUR

T1 - Nonseparable closed vector subspaces of separable topological vector spaces

AU - Ka̧kol, Jerzy

AU - Leiderman, Arkady G.

AU - Morris, Sidney A.

N1 - Funding Information:
The first named author was supported by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, Grant PROMETEO/2013/058 and by the GAČR project I 2374-N35 and RVO: 67985840.
Publisher Copyright:
© 2016, Springer-Verlag Wien.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space [InlineEquation not available: see fulltext.] have a nonseparable closed vector subspace, where c is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space [InlineEquation not available: see fulltext.] has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line M the space of all continuous real-valued functions on M endowed with the pointwise convergence topology, Cp(M) contains a nonseparable closed vector subspace while Cp(M) is separable.

AB - In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space [InlineEquation not available: see fulltext.] have a nonseparable closed vector subspace, where c is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space [InlineEquation not available: see fulltext.] has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line M the space of all continuous real-valued functions on M endowed with the pointwise convergence topology, Cp(M) contains a nonseparable closed vector subspace while Cp(M) is separable.

KW - Locally convex topological vector space

KW - Separable topological space

UR - http://www.scopus.com/inward/record.url?scp=84954460567&partnerID=8YFLogxK

U2 - 10.1007/s00605-016-0876-2

DO - 10.1007/s00605-016-0876-2

M3 - Article

AN - SCOPUS:84954460567

SN - 0026-9255

VL - 182

SP - 39

EP - 47

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 1

ER -