DOI: 10.68381/jca17025 ISSN: 0944-6532
A Unified Construction Yielding Precisely Hilbert and James Sequences Spaces
Dušan Repovš, Pavel V. Semenov
Following R. C. James' approach, we shall define the Banach space
J(e)
J
(
e
)
for each vector
e=(e_1,e_2,...,e_d) \in \Bbb{R}^d
e
=
(
e
1
,
e
2
,
.
.
.
,
e
d
)
∈
R
d
with
e_1 \ne 0
e
1
≠
0
. The construction immediately implies that
J(1)
J
(
1
)
coincides with the Hilbert space
l_2
l
2
and that
J(1;-1)
J
(
1
;
−
1
)
coincides with the celebrated quasireflexive James space
J
J
. The results of this paper show that, up to an isomorphism, there are only these two possibilities: (i)
J(e)
J
(
e
)
is isomorphic to
l_2
l
2
if
e_1+e_2+...+e_d\ne 0
e
1
+
e
2
+
.
.
.
+
e
d
≠
0
, and (ii)
J(e)
J
(
e
)
is isomorphic to
J
J
if
e_1+e_2+...+e_d =0
e
1
+
e
2
+
.
.
.
+
e
d
=
0
. Such a dichotomy also holds for every separable Orlicz sequence space
l_M
l
M
.