Let *X* be a normed space and denote by *X ^{*}* the space of all bounded linear functionals on

*X*. Take a linear subspace

*G ≤ X*which separates the elements of

^{*}*X*, i.e., for each

*x ∈ X*, there is an

*f ∈ G*with

*f(x) ≠ 0*. Denote by

*B*the closed unit ball in

*X*. Now consider a linear subspace

*Y ≤ X*. The question is:

If *Y* is dense in *X* in the weak topology induced by *G*, is *Y ∩ B* necessarily dense in *X ∩ B* in that topology?

## REMARKS, BACKGROUND AND MOTIVATION

Without the assumption that *G* separates points, there exists a trivial counter-example.
Take *X* := ℝ^{2} with the supremum norm, i.e., ∥(*x*, *y*)∥ :=
max{|*x*|, |*y*|}. For *G*, take the linear span of the linear functional *f(x, y) := x + y*. Finally, take *Y* := {(*x*, 0) ; *x* ∈ ℝ}. Then *Y* is *G*-dense in *X* because (*x*, *y*) and (*x* + *y*, 0) are indistinguishable in the *G*-topology. However, the element (1, 1) ∈ *B* is not in the closure of *Y ∩ B* because *f*(1, 1) = 2
and *f(x, 0) ≤ 1* for each *x* with *(x, 0) ∈ B*.

An interesting example is to take the space *G := L ^{∞}(S)*, the space of all bounded measurable functions on a measurable space

*S*, equipped with the supremum norm. Take

*X := G*, with the corresponding dual norm. The space

^{*}*G*can be naturally considered as a subspace of

*X*. Clearly, it separates the points in

^{*}*X*, and the

*G*-topology is exactly the weak *-topology.

An important subspace of

*X*is

*Y := M(S)*, the space of all real measures on

*S*(with finite total variation). If

*S*is large enough, let's say, ℕ, then

*Y*is a proper subspace of

*X*. It is well-known that

*Y*is weakly *-dense in

*X*, but it is also interesting that

*Y*is weakly *-complete by sequences (see Diestel: Sequences and Series in Banach spaces, Springer-Verlag, 1984).

By the Banach-Alaoglu theorem,

*B*is weakly *-compact. One may wonder whether

*X ∩ B*is also weakly *-compact. The answer is no. However, the argument that

*Y*is weakly *-dense in

*X*is insufficient; a sufficient argument is that that

*Y ∩ B*is weakly *-dense in

*X ∩ B*. Though this is not difficult to prove in our particular case, it might by a non-trivial issue in more general cases. If the answer to my initial question is yes, it will be sufficient to only prove that

*Y*is dense in

*X*.

Many thanks in advance for any answer, reference or comment!