Two 1935 questions of Mazur about polynomials in Banach spaces: counter-examples

Abstract

We give various constructions of which the most significant is a continuous scalar-valued 2-polynomial W on the separable Hilbert space l ₂ and an unbounded set R ⊂ l ₂ such that (i) W is bounded on an ε-neighbourhood of R; (ii) W is unbounded on ½R; (iii) W does not factor through any (continuous or not) 1-polynomial mapping from l ₂ to a normed space and sending R to 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.)