Given $X,Y$ be normed spaces over $\mathbb k$ (where $\mathbb k$ could be $=\mathbb C$ or $\mathbb R$). Let $T\in L(X,Y) $ (a continuous and linear operator) then we define $T':Y'\to X'$ by $T'y'=y'T$, it's called the transpose of $T$.
Given a Banach space $X$ a projection from $X$ to a closed subspace $M\le X$ is a continuous linear operator $P:X\to X$ such that $Range(P)=M$ and $P^2=P$ thus $P^2$ is the identity on $M$.
Important: When we say that $P$ is a projection to $M$ we consider $M$ closed for some important reasons, that $P$ is surjective over $M$, and that $P$ is the identity only on $M$ in fact $Ker(I-P)=M$.
Prove that if $X$ is a Banach space, then there exist a projection of norm 1, from $X'''$ to $X'$, when we consider that $X'\subset X'''$ under the canonical injection $Jx' (x'')=x''(x')$.
I have no idea how to attack this problem, I don't know any non-zero map either.