TITLE: Location of a Banach Space inside its bidual
QUESTION [5 upvotes]: Let $X$ be a Banach Space and let $Y$ be a closed subspace of $X^{**}$ such that $X\bigcap Y=0$. Let $P$ be the quotient map from $X^{**}$ onto $X^{**}/ Y$. I need to prove or refute that $P\left|_{X}\right.$ has a closed range (or equivalently is bicontinuous), or is at least a semiembedding (meaning that closed balls are mapped into closed sets).
The case when $X$ is a Frechet Space (not Banach) is also of interest.
I feel I may be overlooking something obvious, and I am sorry if this is the case.
Thank you.

REPLY [6 votes]: I think that $P|_X$ does not have to be a semi-embedding and this can be done as follows. Let $X$ be hereditary $c_0$, $M$ be a total nonnorming subspace in $X$, and $Y=M^{\perp}\subset X^{**}$. Let $P:X^{**}\to X^{**}/Y$ be the quotient map. Since $X^{**}/Y$ can be identified with $M^*$ we get that $||Px||=\sup\{|f(x)|:~ f\in M, ||f||=1\}$. Since $M$ is nonnorming, this map is not an isomorphism, and since $X$ is hereditary $c_0$, the restriction $P|_X$ is not a semi-embedding by L.Drewnowski, [Proc. Edinburgh Math. Soc. (2) 26 (1983), no. 2, 163–167].