We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-9,
author = {Lorenz Halbeisen},
title = {On the complexity of Hamel bases of infinite-dimensional Banach spaces},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {133-134},
zbl = {0998.46008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-9}
}
Lorenz Halbeisen. On the complexity of Hamel bases of infinite-dimensional Banach spaces. Colloquium Mathematicae, Tome 89 (2001) pp. 133-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-1-9/