Abstract
Let
X
be a Banach space,
u\in X^{**}
and
K, Z
two subsets of
X^{**}
. Denote by
d(u,Z)
and
d(K,Z)
the distances to
Z
from the point
u
and from the subset
K
respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w
^*
-compact subset
K\subset X^{**}
such that
d(K,X)=0
satisfies
d(\overline{\text{co}}^{w^*}(K),X)=0
. We extend this result in the following way: if
Z\subset X
is a closed subspace of
X
and
K\subset X^{**}
is a w
^*-
compact subset of
X^{**}
, then
d(\overline{\text{co}}^{w^*}(K),Z)\leq 5 d(K,Z).
Moreover, if
Z\cap K
is w
^*
-dense in
K
, then
d(\overline{\text{co}}^{w^*}(K),Z)\leq 2 d(K,Z)
. However, the equality
d(K,X)=d(\overline{\text{co}}^{w^*}(K),X)
holds in many cases, for instance, if
\ell_1\not\subseteq X^*
, if
X
has w
^*
-angelic dual unit ball (for example, if
X
is WCG or WLD), if
X=\ell_1(I)
, if
K
is fragmented by the norm of
X^{**}
, etc. We also construct under
CH
a w
^*
-compact subset
K\subset B(X^{**})
such that
K\cap X
is w
^*
-dense in
K
,
d(K,X)=\frac 12
and
d(\overline{\text{co}}^{w^*}(K),X)=1
.