Bari bases of subspaces

We shall establish certain characteristic properties of Bari bases of subspaces. We shall show that a complete sequence of finite-dimensional subspaces $\{\mathfrak{R}_j\}_1^\infty$ is a Bari basis if and only if each sequence $\{\psi_j\}_1^\infty$ ($\psi_j\in\mathfrak{R}_j$, $||\psi_j||=1$) is a Bari basis of its own closed linear hull.