Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form (Z^D)*y'(z) = A(z)y(z) + B(z)y(alphaz) + C(z)y'(alphaz) in the neighborhood of the singularity at Z = 0.