Let $X$ be a Banach space. Is $X$ a closed subspace of $X''$?

I'm trying to prove that $X$ is closed in $X''$, where $X$ is a Banach space. I know that $X$ is embeddable in $X''$.

If the isomorphism was bijective, I could show that $X$ is closed since $X''$ closed and closeness is preserved. But in general, it isn't bijective.

Which approach should I take? Hints only please, if possible.

-
Sketch: $X$ is complete. The embedding $(Jx)(\varphi) = \varphi(x)$ is an isometry, so $J(X)$ is complete. Complete subspaces are always closed.