How to show that these spaces are closed and complemented? 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.

Let $f\in X^*$, and $x=x_0+x_1\in X$ (with $x_0=Px$ and hence $x_1=(I-P)x$). Let moreover $f_i:=f|_{X_i}\in (X_i)^*$ for $i=0,1$. Then we have
$$\langle f,x\rangle = f(x)=f(x_0)+f(x_1)= f_0(x_0)+f_1(x_1)= \\
=\langle f_0,Px\rangle + \langle f_1,(I-P)x\rangle = \langle P^*(f_0), x\rangle
+ \langle (I-P)^*(f_1), x\rangle = \\
= \langle P^*(f_0)+(I-P)^*(f_1)\, ,\, x\rangle $$
So, $f=P^*(f_0)+(I-P)^*(f_1)$. You're left to prove closedness and disjointness.
