Norming subspaces of duals of quotient spaces In Davis, William J., and William B. Johnson. "Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces." Israel Journal of Mathematics 14.4 (1973): 353-367., the authors discuss the following problem 
"If $M$ is a norming subspace of $X^*$ and $Y$ is an $M$-closed subspace of $X$, then is $M \cap Y^\perp$ a norming subspace of $(X/Y)^*$ (where $Y^\perp$ is identified with $(X/Y)^*$ in the canonical way)?"
In the paper, the authors conclude that if $X$ is not quasi-reflexive, then there exists a norming subspace $M$ such that the conclusion fails. 
What I am interested in is whether the result can be salvaged in some way by adding some reasonable conditions on $M$, and I would like to know if there is anything in the literature that has addressed this. The papers citing this paper do not discuss this particular question any further.
Thank you for your time. 
 A: Some comments on the problem (I print them as an answer, because they are too lengthy for a comment):
(1) As I understand, the condition that $Y$ is $M$-closed is
equivalent to the condition that $M\cap Y^\perp$ is total over the
quotient $X/Y$.
(2) So, if $X/Y$ is quasireflexive, then "$M\cap Y^\perp$ is
total" implies "$M\cap Y^\perp$ is norming" (we do not need
quasireflexivity of $X$ itself).
(3) You would like to get a condition on $M$ under which for
any subspace $Y$ in $X$ the condition "$M\cap Y^\perp$ is total"
implies "$M\cap Y^\perp$ is norming".
(4) There is a trivial condition of this type: $M$ is of finite
codimension in $X^*$ (but I doubt that it is satisfactory for
you).
(5) It is tempting to claim that any $M$ of infinite codimension
in $X^*$ fails to satisfy this condition for some $Y$, but this is
not true. If $X$ itself is a dual space and $M$ is its predual,
then $M$ is of infinite codimension in $X^*$ for
non-quasireflexive $X$, on the other hand, the condition from (3)
is satisfied. In fact, if $Y$ is $M$-closed, then $X/Y$ is the
dual of $Y^\perp\cap M$, and so  "$Y^\perp\cap M$ is norming over
$X/Y$"
(6) Of course, one can investigate this matter further. My survey Weak* sequential closures in Banach space theory and their applications
could be helpful for this. The survey was written in 1999, but I
did not notice any further work in this direction except for my
note Weak$^*$ closures and derived sets in dual Banach spaces. Possibly I missed something because I do not work in this
direction now.
