A
Banach space
with the property that for all separable Banach spaces
(cf.
Separable space),
every bounded
linear operator
from
to
is
weakly compact
(i.e.,
sends bounded subsets of
into weakly compact subsets of
).
The above property is equivalent to each of the following assertions (see
[a4],
[a5],
[a9]).
1)
Every
weak-
convergent sequence in the dual space
of
is weakly convergent.
2)
Every bounded linear operator
from
to
is weakly compact.
3)
For all Banach spaces
such that
has a
weak-
sequentially compact unit ball, every bounded linear operator
from
to
is weakly compact.
4)
For all
weakly compactly generated Banach spaces
(i.e.,
is the closed linear span of a relatively weakly compact set), every bounded linear
operator from
to
is weakly compact.
5)
For an arbitrary Banach space
,
the limit of any weakly convergent sequence of weakly compact operators from
to
is also a weakly compact operator.
6)
For any Banach space
,
the limit of any strongly convergent sequence of weakly compact operators from
to
is also a weakly compact operator.
Hence, besides the definition given at the beginning, either 1) or 2) can also be
used as the definition of a Grothendieck space. Quotient spaces and complemented
subspaces of a Grothendieck space are also Grothendieck spaces.
Reflexive Banach spaces are obvious examples of Grothendieck spaces (cf.
Reflexive space).
Every separable quotient space of a Grothendieck space is necessarily reflexive.
The first non-trivial example of a Grothendieck space is the space
of continuous functions on a
compact Stonean space
(i.e., a compact
Hausdorff space
in which each open set has an open closure)
[a6].
Other examples of Grothendieck spaces are:
,
where
is a
compact
-Stonean space
(each open
-set
has an open closure) or a
compact
-space
(any two disjoint open
-sets
have disjoint closures) (see
[a1],
[a10]);
,
where
is a positive measure;
,
where
is a
-algebra
of subsets of
;
injective Banach spaces; the Hardy space
of all bounded analytic functions on the open unit disc
[a2];
and
von Neumann algebras
[a8].
A uniformly bounded
-semi-group
of operators (cf.
Semi-group of operators)
on a Grothendieck space is
strongly ergodic
if and only if the
weak-
closure and the strong closure of the range of the dual operator of the generator
coincide
[a11].
If
is a Grothendieck space, then every sequence
of contractions on
which converges to the identity in the strong operator topology actually converges
in the uniform operator topology (see
[a3],
[a7]).
In particular, this implies equivalence of strong continuity and uniform
continuity for contraction
-semi-groups
on
.