Smooth 2-homogeneous polynomials on Hilbert spaces

Abstract.

We determine the smooth points of the unit ball of the space of 2-homogeneous polynomials on a Hilbert space H. Working separately for the real and the complex cases we show that a smooth polynomial attains its norm. We deduce that the polynomial P is smooth if and only if there exists a unit vector x ₀ in H such that $P(x)=\pm \left \langle x,x_{0}\right \rangle ^{2}+P_{1}(x_{1})$ where $x=\left \langle x,x_{0}\right \rangle x_{0}+x_{1}$ is the decomposition of x in $H={\rm {span}}\{ x_{0}\} \oplus H_{1}$ and P ₁ is a 2-homogeneous polynomial on H ₁ of norm strictly less than 1.