Suppose $X$ is a Banach space and $X_0$,$X_1$ are closed complemented subspaces of $X$. Let $P:X\rightarrow X_0$ and $I-P:X\rightarrow X_1$ ($I$ is the identity operator) be the projections associated with the decomposition $X=X_0\oplus X_1$.
Define $P^\star:X_0^\star\rightarrow X^\star$ $$\langle P^\star (x),y\rangle=\langle x,P(y)\rangle$$
In the same way, define $(I-P)^\star:X_1^\star\rightarrow X^\star$. How can one show that $P^\star(X_0^\star)$ and $(I-P)^\star(X_1^\star)$ are closed complemented subspaces of $X^\star$?
Note: $\star$ denotes dual and $\langle\cdot,\cdot\rangle$ duality.