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⊂
l
2
such that (i)
W
is bounded on an
ϵ
-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 $\R$ and $\C$. (In finite dimensions the questions were answered in the positive by Auerbach soon after being asked.)