Fourier transform of function on locally compact Abelian groups taking   value in Banach spaces
  We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded operator. If $G$ is an infinite group then Fourier transform $F: L_2(G,X)\to L_2(\widehat G,X)$ is a bounded operator if and only if Banach space $X$ is isomorphic to a Hilbert one. 
Yauhen Radyna, Anna Sidorik