Quadratic exponential vectors

We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L(2)(R(d))boolean AND L(infinity)(R(d)). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)], we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.