ON QUADRATIC FUNCTIONALS AND SOME PROPERTIES OF HAMEL BASES

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. © 1990.