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

Abstract

We construct a continuous scalar-valued 2-polynomial, $W$, on the separable Hilbert space $l_2$ and an unbounded set $R\subset l_2$ such that (i) $W$ is bounded on an $ε$-neighbourhood of $R$; (ii) $W$ is unbounded on ${1/2} R$; (iii) consequently, $W$ does not factor through any bounded 1-polynomial on $l_2$ 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 $\R$ and $\C$. (In finite dimensions the questions were answered in the positive by Auerbach soon after being asked.)