Suppose $X$ and $Y$ are Banach spaces. If $X$ are $Y$ are isomorphic and $X$ is complemented in $X^{**}$, then $Y$ is complemented in $Y^{**}$.
I am self-studying the problem $1.2$ from here.
$A$ is complemented in $B$ if there exists a projection $P : B \rightarrow A$ such that $P$ is onto.
I want to show that there is a projection from $Y^{**}$ to $Y$, but how to obtain such projection?
Let $P : X^{**} \to X^{**}$ be a continuous projection whose range is $X$. If $C : X \to Y$ is a bijection, then its bi-adjoint $C^{**} : X^{**} \to Y^{**}$ is a bijection. Now you can show that $\tilde P := C^{**} P C^{-**}$ is a continuous projection and the range of $\tilde P$ is $Y$.