Given a stochastic differential equation with path-dependent coefficients
driven by a multidimensional Wiener process, we show that the support of the
law of the solution is given by the image of the Cameron-Martin space under the
flow of the solutions of a system of path-dependent (ordinary) differential
equations. Our result extends the Stroock-Varadhan support theorem for
diffusion processes to the case of SDEs with path-dependent coefficients. The
proof is based on the Functional Ito calculus and interpolation estimates for
stochastic processes in Holder norm.