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.



Location of a Banach Space inside its bidual
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]. 


