We present a new way of obtaining the Bargmann transform between L 2 (R n) and the Fock space F = F(C n) via a simple restriction principle applied to holo-morphic functions. This same principle also recovers the transform between functions on a compact Lie group and holomorphic functions on its complexii-cation studied by Gross, Hall, Hijab et al., see 1] and 2], and it gives in a similar way ...