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 $\epsilon$-neighbourhood of $R$; (ii) $W$ is unbounded on $1/2R$; (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 $\mathbb{R}$ and $\C$. (In finite dimensions the questions were answered in the positive by Auerbach soon after being asked.)

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