It is shown that a separable Banach space
X
X
can be given an equivalent norm
|
|
|
⋅
|
|
|
|||\, \cdot |||\,
with the following properties: If
(
x
n
)
⊆
X
(x_{n})\subseteq X
is relatively weakly compact and
lim
m
→
∞
lim
n
→
∞
|
|
|
x
m
+
x
n
|
|
|
=
2
lim
m
→
∞
|
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|
x
m
|
|
|
\lim _{m\to \infty } \lim _{n\to \infty } |||\, x_{m}+x_{n}|||\, = 2\lim _{m\to \infty } |||\, x_{m}|||\,
, then
(
x
n
)
(x_{n})
converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in
(
X
,
|
|
|
⋅
|
|
|
)
(X,|||\, \cdot |||\, )
is 1-equivalent to the unit vector basis of
ℓ
1
\ell _{1}
(respectively,
c
0
c_{0}
) implies that
X
X
contains an isomorph of
ℓ
1
\ell _{1}
(respectively,
c
0
c_{0}
).