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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.

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1 Answer 1

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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.

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Please verify if I am right. I proved that $P^\star$ onto $P^\star(X_0^\star)$ is an isometry. Hence, prove that $P^\star(X_0^\star)$ is closed, is the same as to prove that $X_0^\star$ is closed. –  Tomás Feb 7 '13 at 13:18
    
Or better saying, I think I have to prove that $X_0^\star$ is complete. Is this right? –  Tomás Feb 7 '13 at 13:36

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