Question

Is a Banach space weak-* dense in its quadruple dual?

Answer 1

Let v in X** so that dist(v, i(X)) = 1 - this exists because i(X) is closed in X\*\* but not all of X\*\*. Then by Hahn-Banach there is a functional f: X\*\* -> k so that f(v) = 1 and f(x) = 0 for all x in i(E).

Now to check that j(v) cannot be weak* approximated by elements of the form j(i(x)) it is enough to see that

j(v) (f) = 1

while

j(i(x)) (f) = 0

for all x in X.