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Smooth 2-homogeneous polynomials on Hilbert spaces
Authors:B C Grecu
Institution:Mathematics Department, National University of Ireland, Galway, Ireland, IE
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 x0 in H such that P(x)=± á x,x0 ñ 2+P1(x1)P(x)=\pm \left \langle x,x_{0}\right \rangle ^{2}+P_{1}(x_{1}) where x= á x,x0 ñ x0+x1 x=\left \langle x,x_{0}\right \rangle x_{0}+x_{1} is the decomposition of x in H=span{ x0} ?H1H={\rm {span}}\{ x_{0}\} \oplus H_{1} and P1 is a 2-homogeneous polynomial on H1 of norm strictly less than 1.
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