This paper is concerned with the existence of solutions to the initial value problem for a semilinear functional differential equation with infinite delay in a Banach space:
$$\begin{cases}x^\prime(t)=Lx_t+f(t,x_t),\quad t\in[0,a]\ (a > 0)\\ x(t)=\varphi(t),\quad t\in ]-\infty,0].\end{cases}$$The result is based on the Leray-Schauder alternative and a priori bounds on solutions.