On the complexity of Hamel bases of infinite dimensional Banach spaces

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 will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko

A result related to the problem CN of Fremlin

We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of second category in the box product topology