On quadratic functionals and some properties of Hamel bases

Abstract

It is shown that any quadratic functional q : R → R whose restriction to a second category Baire subset of R is continuous must be continuous on the whole real line. For this purpose we investigate regularity properties of quadratic functionals with a continuous restriction on an analytic set containing 1 2 (H + H) , where H is a Hamel basis of R . We also prove that for any Baire set T ⊂ R of the second category there exists a Hamel basis H ⊂ R such that 1 2 (H + H) ⊂ T .