In this paper, we define generalized Fourier-Hermite functionals on a function space $C_{a,b}[0,;T]$ to obtain a complete orthonormal set in $L_2(C_{a,b}[0,;T])$ where $C_{a,b}[0,;T]$ is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in $L_2(C_{a,b}[0,;T])$ has a generalized Fourier-Wiener function space transform ${cal{F}}_{sqrt{2},i}(F)$ also belonging to $L_2(C_{a,b}[0,;T])$.