Let $X$ be a normed space and $X^{**}$is its bidual.
I’m trying to show $X \subseteq X^{**}$.
Let $J:X \to X^{**} $ be canonical embedding. Since it is an isometry (I’ve shown that) it is invertible.
$x \in X$ then $J(x) \in X^{**}$ thus $J^{-1}(J(x))=x$. How can I say $J^{-1}(J(x)) \in X^{**}$?
I think I’m writing unnecessary things. I couldn’t show $x \in X^{**}$. It is probably so easy but I’m stuck.
I’m sorry in advance if there is any mistake.
Thanks for any help.
$X \subseteq X^{\ast\ast}$ is misleading: It's rather $J[X] \subseteq X^{\ast\ast}$, which is isometrically isomorphic to $X$, via $J$, so $\|J(x)\|=\|x\|$ (in the respective norms on $X^{\ast\ast}$ and $X$).
So we can conceptually see $J(x)$ (i.e. all point evaluations) as a "copy" of $X$ inside $X^{\ast\ast}$.
So no need to mess with $J^{-1}(J(x))$, that's just $x$ by definition.