We give various constructions of which the most significant is a continuous scalar-valued 2-polynomialWon the separable Hilbert spacel2and an unbounded setR⊂l2such that (i)Wis bounded on anε-neighbourhood ofR; (ii)Wis unbounded on ½R; (iii)Wdoes not factor through any (continuous or not) 1-polynomial mapping froml2to a normed space and sendingRto a bounded set. This answers in the negative two 1935 questions asked by Mazur (Problems 55 and 75 in the Scottish Book). The construction is valid both over the real and complex fields. (In finite dimensions, the questions were answered in the positive by Auerbach soon after being asked.)