IJMMS 27:5 (2001) 289–300
PII. S0161171201005683
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
SOME PROPERTIES OF BANACH-VALUED
SEQUENCE SPACES p [X]
QINGYING BU
(Received 7 August 2000)
Abstract. We discuss some properties of the Banach-valued sequence space p [X] (1 ≤
p < ∞), the space of weakly p-summable sequences on a Banach space X. For example,
we characterize the reflexivity of p [X], convergent sequences on p [X], and compact
subsets of p [X].
2000 Mathematics Subject Classification. 40A05, 46B45.
1. Introduction. It is known that the general theory of scalar-valued sequence spaces (SVSS) plays an important role in the theory of topological vector spaces (cf.
[11, 21]). Furthermore, the theory of generalized sequence spaces, or vector-valued
sequence spaces (VVSS) (cf. [10, 18, 24]) which has emerged as an outgrowth of the
development of SVSS also plays an important role in the theory of locally convex
spaces, especially in the investigation of nuclear spaces through λ-summing operators (cf. [3, 4, 5, 19, 20]). For example, Pietsch [19, 20] characterized a nuclear locally
convex space X in terms of the absolutely p-summable sequence space p (X) and the
weakly p-summable sequence space p [X]. Because of Pietsch’s work, people began to
be interested in the properties of the spaces p (X) and p [X]. Some results about the
space p (X) are presented (cf. [1, 2, 6, 7, 8, 12, 14, 15, 16, 17]). But few results about
the space p [X] have been presented for a long time. Recently, Wu and Bu [26] have
shown a representation of the Köthe dual of the space p [X] and a characterization
that p [X] is a Grothendieck space. Gupta and Bu [9] have characterized GAK-ness of
the space p [X] in terms of the (q)-property of a Banach space X for 1 < p < ∞, and
in terms of non-containment of c0 of X for p = 1 (a locally convex space containing
no copy of c0 has a number of nice properties, cf. [13, 25]).
In Section 3, by the definitions and basic results given in [9, 26], we characterize the
GAK-ness of the space p [X] in terms of the Köthe dual p [X]× and the topological
dual p [X]∗ of p [X]. Then by using this result, we characterize the reflexivity of the
space p [X] in terms of the reflexivity and (q)-property of a Banach space X. This
result shows that the characterization of the reflexivity of the space p [X] is different
from the characterization of the reflexivity of the space p (X) (the reflexivity of p (X)
is discussed in [12]).
In Section 4, by using the characterization that p [X] is a GAK-space (cf. [9]) and
some properties about Köthe dual of p [X] (cf. [26]), we characterize convergent
sequences of p [X] with respect to the norm topology and with respect to the weak
topology.
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QINGYING BU
In Section 5, by using the results in Section 4, we characterize compact subsets of
p [X] and relatively weakly sequentially compact subsets of p [X].
2. Concepts and basic results. Throughout this paper, we denote a Banach space
by X and its topological dual by X ∗ . Then (X, X ∗ ) forms a dual pair (cf. [23]). We
denote the strong topology and the weak topology with respect to the dual pair (X, X ∗ )
by β(X, X ∗ ) and σ (X, X ∗ ), respectively. BX stands for the closed unit ball of X. For
1 ≤ p < ∞, we introduce the Banach-valued sequence space p [X], the space of weakly
p-summable sequences on X, that is,
p
f xi < ∞ ∀f ∈ X ∗
p [X] = x̄ = xi i ∈ X :
N
(2.1)
i≥1
and introduce a norm · p on p [X], that is,


1/p
 
p
f xi 
: f ∈ BX ∗ .
x̄p = sup 


(2.2)
i≥1
Then (p [X], ·p ) is a Banach space (cf. [9, 26]). By the results in [9], we have another
form about p [X] and · p as follows.
Proposition 2.1. The space of weekly p-summable sequences on X has the form
p [X] = x̄ = xi
i
N
∈X :
ti xi converges ∀ ti i ∈ q .
(2.3)
i≥1
and for each x̄ ∈ p [X],




ti xi : ti i ∈ Bq .
x̄p = sup 

i≥1
(2.4)
Here the space q , denotes the dual space of p for 1 < p < ∞ (i.e., 1/p + 1/q = 1), and
denotes the space c0 for p = 1.
It is easy to see that the coordinate projections
Pi : p [X] → X,
Pi (x̄) = xi
for i = 1, 2, . . .
(2.5)
are continuous. For x̄ ∈ X N , we introduce the symbols
x̄(i ≤ n) = x1 , x2 , . . . , xn , 0, 0, . . . ;
x̄(i > n) = 0, . . . , 0, xn+1 , xn+2 , . . . .
(2.6)
Definition 2.2. The Banach-valued sequence space p [X] is called a GAK-space if
for all x̄ ∈ p [X], limn x̄(i > n)p = 0 (cf. [9, 10, 26]).
We denote the GAK-subspace of p [X] by p [X]G , that is,
p [X]G = x̄ ∈ p [X] : lim x̄(i > n)p = 0 .
n
(2.7)
SOME PROPERTIES OF BANACH-VALUED SEQUENCE SPACES p [X]
291
The Köthe dual of p [X] with respect to the dual pair (X, X ∗ ) is denoted by
p [X]× |(X,X ∗ ) (cf. [10, 26]), that is,
∗N
¯
fi xi < ∞ ∀x̄ ∈ p [X] .
p [X] |(X,X ∗ ) = f = fi i ∈ X :
×
(2.8)
i≥1
We denote p [X]× |(X,X ∗ ) simply by p [X]× if the meaning is clear from the context.
Proposition 2.3. The Köthe dual of p [X]× is
p [X]× |(X,X ∗ )
×
|(X ∗ ,X ∗∗ ) = p X ∗∗ .
(2.9)
Proof. See [26].
For x̄ ∈ p [X] and f¯ ∈ p [X]× , define
fi xi .
x̄, f¯ =
(2.10)
i≥1
Then (p [X], p [X]× ) forms a dual pair with respect to the bilinear functional ·, ·
defined in (2.10). And for each f¯ ∈ p [X]× , ·, f¯
is a linear functional on p [X]. Furthermore, we have the following proposition.
Proposition 2.4. For each f¯ ∈ p [X]× , ·, f¯
is a continuous linear functional on
(p [X], · p ), that is,
∗
·, f¯ ∈ p [X], · p := p [X]∗ .
(2.11)
Proof. For f¯ ∈ p [X]× , n ∈ N, define the linear functionals F and Fn on p [X] by
F (x̄) =
fi xi ,
Fn (x̄) =
i≥1
n
fi xi .
(2.12)
i=1
Then it is easy to see that Fn ∈ p [X]∗ and for each x̄ ∈ p [X], limn Fn (x̄) = F (x̄). So
the Banach-Steinhaus theorem (cf. [23, page 137]) implies that F ∈ p [X]∗ .
By Proposition 2.4, the space p [X]× can be considered as a subspace of p [X]∗ ,
that is,
p [X]× ⊆ p [X]∗ .
(2.13)
Now denote the dual norm of · p on the dual space p [X]∗ by · ∗
p . Then we
have the following proposition belonging to [26].
Proposition 2.5. For 1 < p < ∞, (p [X]× , · ∗
p ) is a GAK-space.
By [26, Lemma 1] and [10, Corollary 4.9], we have the following two propositions.
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QINGYING BU
Proposition 2.6. For 1 ≤ p < ∞,
× ∗
∗
p [X]× = p [X]G = p [X]G , · p := p [X]G .
(2.14)
Proposition 2.7. If a Banach-valued sequence space (λ[X], · ) is both a Banach
space and a GAK-space, then (λ[X], · )∗ = λ[X]× |(X,X ∗ ) .
Definition 2.8. For 1 < q ≤ ∞, a Banach space X is said to have the (q)-property
if the following two statements about a sequence {xi }∞
1 in X are equivalent:
t
x
converges
for
each
(t
)
∈
;
(i)
i i
q
i≥1 i i
(ii)
i≥1 ti xi converges uniformly for all (ti )i ∈ Bq .
Swartz [22] has proved that every Banach space has the (∞)-property. More results
about (q)-property can be seen in [9]. For 1 < p < ∞, let q be its conjugate, that is,
1/p + 1/q = 1. Then we have the following proposition which belongs to [9].
Proposition 2.9. The space p [X] (1 < p < ∞) is a GAK-space if and only if X has
the (q)-property.
3. Reflexivity
Lemma 3.1. For a sequence x̄ ∈ p [X], define a linear map
Ix̄ : p [X]× → 1 ,
Ix̄ f¯ = fi xi i .
(3.1)
Then Ix̄ is σ (p [X]× , p [X])-σ (1 , ∞ ) continuous.
Proof. The proof can be easily completed and so is omitted.
(n)
×
Lemma 3.2. Let f¯(n) = (fi )i ∈ p [X]× such that {f¯(n) }∞
1 is σ (p [X] , p [X])(n)
bounded. If for each i ∈ N, there exists fi in X ∗ such that σ (X ∗ , X)- limn fi
f¯ = (fi )i ∈ p [X]× .
= fi , then
Proof. Let x̄ ∈ p [X]. By Lemma 3.1, {(fi (xi ))i }∞
n=1 is a σ (1 , ∞ )-bounded subset of 1 and hence, is β(1 , ∞ )-bounded. Thus
(n)
M = sup
(n) fi
xi < ∞.
(3.2)
n≥1 i≥1
(n)
Fix m ∈ N. Since σ (X ∗ , X)- limn fi = fi for each i ∈ N, there exists an n ∈ N such
that
1
(n) fi
, i = 1, 2, . . . , m.
(3.3)
xi − fi xi <
m
So
m
m m (n) (n) fi xi ≤
f
f
xi − fi xi +
xi < 1 + M.
i
i=1
Since m is arbitrary in N,
i=1
i≥1 |fi (xi )| < ∞.
i
(3.4)
i=1
Thus we have proved that f¯ ∈ p [X]× .
Theorem 3.3. For 1 ≤ p < ∞, the following hold:
(i) p [X]× is a closed subspace of p [X]∗ ;
SOME PROPERTIES OF BANACH-VALUED SEQUENCE SPACES p [X]
293
(ii) for each x̄ ∈ p [X],
∗
x̄p = sup x̄, f¯ : f¯ ∈ p [X]× , f¯ ≤ 1 ;
p
(3.5)
(iii) p [X]× = p [X]∗ if and only if p [X] is a GAK-space.
Proof. (i) To prove that p [X]× is a closed subspace of p [X]∗ , we only need to
prove that p [X]× is complete with respect to the norm · ∗
p.
(n) ∞
×
∗
¯
Let {f } be a Cauchy sequence in (p [X] , · ). By the continuity of the coor1
p
∗
∗
}∞
dinate projections from (p [X]× , · ∗
n=1 is a Cauchy sequence in X
p ) to X , {fi
for each i ∈ N. Thus the completeness of X ∗ implies that there exist fi in X ∗ such
(n)
that limn fi = fi for each i ∈ N. So f¯ = (fi )i ∈ p [X]× by Lemma 3.2. Next we will
prove that limn f¯(n) = f¯.
(n)
Fix x̄ ∈ p [X]. By Lemma 3.1, {(fi (xi ))i }∞
n=1 is a σ (1 , ∞ )-Cauchy sequence in
1 and hence, is β(1 , ∞ )-Cauchy. So for any given ε > 0, there exists an n0 ∈ N such
that for n > n0 ,
(n) ε
(n ) fi
(3.6)
xi − fi 0 xi < .
4
i≥1
(n)
Since f¯, f¯(n0 ) ∈ p [X]× , there exists an i0 ∈ N such that
ε
fi xi < ,
4
i>i
0
(n)
And furthermore, since limn fi
n1 > n0 such that for n > n1 ,
(n0 ) ε
fi
xi < .
4
i>i
(3.7)
0
= fi for each i ∈ N, there exists an n1 ∈ N with
ε
(n) fi
,
xi − fi xi <
4i0
i = 1, 2, . . . , i0 .
(3.8)
So for n > n1 , we have
i0 (n) (n) (n ) fi
fi
x̄, f¯(n) − f¯ ≤
xi − fi xi +
xi − fi 0 xi i=1
i>i0
(n0 ) fi xi < ε.
fi
xi +
+
i>i0
(3.9)
i>i0
¯(n) = f¯.
It follows that weak∗ - limn f¯(n) = f¯. Note that {f¯(n) }∞
1 is Cauchy. So limn f
×
∗
Therefore, we have proved that (p [X] , · p ) is complete and (i) follows.
(ii) Let x̄ ∈ p [X] and q the conjugate of p, that is, 1/p + 1/q = 1. Then


1/p
 p



f xi : f ∈ BX ∗
x̄p = sup


i≥1
∗
ti f xi : f ∈ BX , ti i ∈ Bq
= sup i≥1
294
QINGYING BU
= sup x̄, ti f i : f ∈ BX ∗ , ti i ∈ Bq
∗
≤ sup x̄, f¯ : f¯ ∈ p [X]× , f¯ ≤ 1
p
(3.10)
since (ti f )i ∈ p [X]× and (ti f )i ∗
p ≤ 1 for each (ti )i ∈ Bq and for each f ∈ BX ∗ . On
the other hand,
x̄p = sup x̄, F : F ∈ p [X]∗ , F ∗
p ≤1
∗
≥ sup x̄, f¯ : f¯ ∈ p [X]× , f¯ ≤ 1 .
(3.11)
p
So (ii) follows.
(iii) If p [X] is a GAK-space, it follows from Proposition 2.6 that p [X]× = p [X]∗ .
On the other hand, suppose that p [X]× = p [X]∗ . Fix x̄ ∈ p [X]. By Lemma 3.1 and
the Banach-Alaoglu theorem (cf. [23, page 130]),
∗
Ix̄ f¯ : f¯ ∈ p [X]× , f¯ ≤ 1
p
(3.12)
is a σ (1 , ∞ )-compact subset of 1 . So by [11, Proposition 6.11, page 108],



∗

× ¯
¯
fi xi : f ∈ p [X] , f ≤ 1 = 0.
lim sup 

p
n
i>n
(3.13)
It follows from (ii) that limn x̄(i > n)p = 0. Thus we have proved that p [X] is a
GAK-space and hence, (iii) follows.
Lemma 3.4. Let (X, · ) be a reflexive Banach space and Y a closed subspace of X.
If (Y , · )∗ = (X, · )∗ , then Y = X.
Proof. The proof can be easily completed and so is omitted.
Theorem 3.5. The Banach space p [X] (1 < p < ∞) is a reflexive space if and only if
(i) X is a reflexive space; and
(ii) p [X] is a GAK-space.
Proof. Suppose that (i) and (ii) hold. By Theorem 3.3, p [X]× = p [X]∗ . Moreover,
by Propositions 2.5 and 2.7,
× ∗ ∗
p [X]× (X ∗ ,X ∗∗ ) = p [X]× , · ∗
= p [X]∗ , · ∗
:= p [X]∗∗ .
p
p
(3.14)
So by Proposition 2.3, p [X ∗∗ ] = p [X]∗∗ . It follows from (i) that p [X] = p [X]∗∗ .
So p [X] is a reflexive space.
On the other hand, suppose that p [X] is a reflexive space. Since X is isometrically
isomorphic to a subspace of p [X], (i) holds.
Now, by Propositions 2.3, 2.5, and 2.7,
p [X]× , · ∗
p
∗
× = p [X]× (X ∗ ,X ∗∗ ) = p X ∗∗ .
(3.15)
SOME PROPERTIES OF BANACH-VALUED SEQUENCE SPACES p [X]
295
Since X and p [X] are reflexive,
p [X]× , · ∗
p
∗
∗
= p X ∗∗ = p [X] = p [X]∗ , · ∗
.
p
(3.16)
It follows from Lemma 3.4 that p [X]× = p [X]∗ . Thus (ii) holds by Theorem 3.3.
Now by Proposition 2.9, we have the following theorem.
Theorem 3.6. The Banach space p [X] is a reflexive space if and only if the Banach
space X is both a reflexive space and has the (q)-property (1 < p < ∞, 1/p + 1/q = 1).
4. Convergent sequences
Lemma 4.1. Let x̄ ∈ p [X] (1 ≤ p < ∞). Then limn x̄(i ≤ n)p = x̄p .
Proof. By Proposition 2.1,




ti xi : ti i ∈ Bq ,
x̄p = sup 

i≥1


n


x̄(i ≤ n) = sup ti xi : ti i ∈ Bq .
p

i=1
(4.1)
It is easy to see that x̄(i ≤ n)p ≤ x̄(i ≤ n+1)p . So {x̄(i ≤ n)p }∞
n=1 is an increasing
sequence. Note that supn≥1 x̄(i ≤ n)p = x̄p . Hence limn x̄(i ≤ n)p = x̄p .
Theorem 4.2. Let x̄ (n) , x̄ ∈ p [X] (1 ≤ p < ∞). Then
lim x̄ (n) = x̄
n
(4.2)
is equivalent to
(n)
(i) limn xi = xi for each i ∈ N; and
(ii) limn x̄ (n) (i > m)p = x̄(i > m)p for each m ∈ N, if and only if p [X] is a
GAK-space.
Proof. (⇐) Suppose that limn x̄ (n) = x̄. Then (i) and (ii) hold obviously from the
continuity of the coordinate projections Pi and the norm · p . On the other hand,
suppose that (i) and (ii) hold. We will prove that limn x̄ (n) = x̄.
Fix ε > 0. Since p [X] is a GAK-space, there exists an m0 ∈ N such that
x̄ i > m0 < ε .
p
4
(4.3)
By (i) and (ii), there exists n0 ∈ N such that for n > n0 ,
(n)
xi − xi <
ε
, for i = 1, 2, . . . , m0 ,
4m0
(n) ε
x̄
i > m0 p − x̄ i > m0 p < .
4
(4.4)
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QINGYING BU
So for n > n0 , we have
(n)
x̄ − x̄ = x̄ (n) i ≤ m0 + x̄ (n) i > m0 − x̄ i ≤ m0 − x̄ i > m0 p
p
≤ x̄ (n) i ≤ m0 − x̄ i ≤ m0 p + x̄ (n) i > m0 p + x̄ i > m0 p
(4.5)
(n)
xi −xi +x̄ (n) i > m0 p −x̄ i > m0 p+2x̄ i > m0 p
m0
≤
i=1
< ε.
Thus we have proved that limn x̄ (n) = x̄ and the sufficiency is proved.
(⇒) For any given x̄ ∈ p [X], we want to show that
lim x̄(i > n)p = 0.
n
(4.6)
(n)
Let x̄ (n) = x̄(i ≤ n). Then for each i ∈ N, xi = xi if n > i. So x̄ (n) and x̄ satisfy
condition (i). Now for each m ∈ N, by Lemma 4.1, limn x̄ (n) (i > m)p = limn x̄(m <
i ≤ n)p = x̄(i > m)p . So x̄ (n) and x̄ satisfy condition (ii). Thus (i) and (ii) imply
that limn x̄ (n) − x̄p = 0. Hence
lim x̄(i > n)p = lim x̄ (n) − x̄ p = 0.
n
n
(4.7)
Therefore, we have proved that p [X] is a GAK-space and the necessity is proved.
Corollary 4.3. Let x̄ (n) , x̄ ∈ p [X]G (1 ≤ p < ∞). Then
lim x̄ (n) = x̄
n
(4.8)
if and only if
(n)
(i) limn xi = xi for each i ∈ N; and
(ii) limn x̄ (n) (i > m)p = x̄(i > m)p for each m ∈ N.
Theorem 4.4. Let x̄ (n) , x̄ ∈ p [X] (1 < p < ∞). Then
σ p [X], p [X]× - lim x̄ (n) = x̄
n
(4.9)
if and only if
(n)
(i) σ (X, X ∗ )- limn xi = xi for each i ∈ N; and
(n)
(ii) supn≥1 x̄ p < ∞.
Proof. It is easy to see that σ (p [X], p [X]× )- limn x̄ (n) = x̄ implies that (i) and
(ii) hold. Conversely, suppose that (i) and (ii) hold. We will prove that
σ p [X], p [X]× - lim x̄ (n) = x̄.
n
(4.10)
Fix any given f¯ ∈ p [X]× and any given ε > 0. Let M = supn≥1 x̄ (n) p . By
Proposition 2.5, p [X]× is a GAK-space. So there exists a k ∈ N such that
ε
f¯(i > k)∗ < .
p
2 M + x̄p
(4.11)
SOME PROPERTIES OF BANACH-VALUED SEQUENCE SPACES p [X]
297
By (i), there exists n0 ∈ N such that for n > n0 ,
ε
fi xi(n) − fi xi <
,
2k
i = 1, 2, . . . , k.
(4.12)
So for n > n0 , we have
k (n)
(n)
(n)
x̄ − x̄, f¯ ≤
fi xi
fi xi − xi − fi xi + i>k
i=1
k fi xi(n) − fi xi + x̄ (n) − x̄, f¯(i > k) =
i=1
(4.13)
∗
ε ≤ + x̄ (n) − x̄ p f¯(i > k)
p
2
∗
ε ≤ + M + x̄p f¯(i > k)p < ε.
2
Thus we have proved that
σ p [X], p [X]× - lim x̄ (n) = x̄.
(4.14)
n
This completes the proof.
Remark 4.5. Theorem 4.4 is not valid for the space 1 [X]. For example, pick x0 ∈ X
and f0 ∈ X ∗ such that f0 (x0 ) = 1. Let

f¯ = f0 , f0 , . . . ,
x̄
(n)

n
! "# $ 1
1
1

= 0, . . . , 0, x0 , 2 x0 , 3 x0 , . . . ,
2
2
2
(4.15)
where (1/2k )x0 is at the (n+k)th place, k = 1, 2, . . . . Then f¯ ∈ 1 [X]× , x̄ (n) ∈ 1 [X] and


 1  (n) x̄ = sup
k f x0 : f ∈ BX ∗ = x0 ,
1

2

n = 1, 2, . . . .
(4.16)
k≥1
So x̄ (n) and 0 satisfy conditions (i) and (ii) in Theorem 4.4. But
1 x̄ (n) , f¯ =
f0 x0 = 1.
2k
k≥1
Hence
σ 1 [X], 1 [X]× - lim x̄ (n) ≠ 0.
n
(4.17)
(4.18)
Thus we have shown that Theorem 4.4 is not valid for the space 1 [X].
5. Compact sets. A subset B of p [X] is called normal if (xi )i ∈ B and (ti )i ∈ ∞
imply that (ti xi )i ∈ B.
Theorem 5.1. A normal subset B of p [X] (1 ≤ p < ∞) is compact if and only if
(i) B is closed;
298
QINGYING BU
(ii) limn x̄(i > n) = 0 uniformly for all x̄ ∈ B;
(iii) Pi (B) is a compact subset of X for each i ∈ N.
Proof. First, let B be a normal compact subset of p [X]. Then (i) holds obviously.
By the continuity of Pi , (iii) holds. Now suppose (ii) does not hold. Then there exist
ε0 > 0, x̄ (k) ∈ B, and n1 < n2 < · · · such that
(k) x̄
i > n k p ≥ ε 0 ,
Let
k = 1, 2, . . . .
(5.1)
(k)
(k)
ȳ (k) = 0, . . . , 0, xnk +1 , xnk +2 , . . . .
(5.2)
Since B is normal, ȳ (k) ∈ B for k = 1, 2, . . . . Noticing that B is compact and ȳ (k) converges to 0 coordinate-wise as k → ∞, limk ȳ (k) = 0 which contradicts (5.1). This contradiction shows that (ii) holds.
Now, suppose (i), (ii), and (iii) hold. We want to show that B is compact. By (i), B is
complete. By [23, page 88], it suffices to show that B is totally bounded.
Fix ε > 0, by (ii), there exists n0 ∈ N such that
ε
sup x̄ i > n0 p : x̄ ∈ B < .
(5.3)
2
%n0
Pi (B) is a totally bounded subset of X. So there exists a finite subset F
By (iii), A =: i=1
of X such that for each x ∈ A, there exists y ∈ F satisfying x − y < ε/2n0 . Now let
D =: ȳ = yi i ∈ X N : yi ∈ F for 1 ≤ i ≤ n0 , yi = 0 for i > n0 .
(5.4)
Then D is a finite subset of p [X]. Now for each x̄ = (xi )i ∈ B, since xi ∈ A for
i = 1, 2, . . . , n0 , there exists yi ∈ F such that
xi − yi <
ε
,
2n0
i = 1, 2, . . . , n0 .
(5.5)
Let ȳ = (y1 , . . . , yn0 , 0, 0, . . .). Then ȳ ∈ D and
x̄ − ȳ ≤ x̄ i ≤ n0 − ȳ i ≤ n0 + x̄ i > n0 p
p
p


n
0
  ε
≤ sup ti xi − yi : ti i ∈ Bq  + 2 < ε,

i=1
(5.6)
where 1/p +1/q = 1. So we have showed that B is totally bounded. This completes the
proof.
×
Lemma 5.2. Let x̄ (n) = (xi )i ∈ p [X] such that {x̄ (n) }∞
1 is σ (p [X], p [X] )(n)
(n)
bounded. If for each i ∈ N, there exists xi ∈ X such that σ (X, X ∗ )- limn xi = xi ,
then x̄ = (xi )i ∈ p [X].
Proof. As in the proof of Lemma 3.2, we can show that i≥1 |fi (xi )| < ∞ for each
f¯ = (fi )i ∈ p [X]× . So x̄ ∈ (p [X]× |(X,X ∗ ) )× |(X ∗ ,X) . By Proposition 2.3,
p [X]× |(X,X ∗ )
× (X ∗ ,X)
Thus x̄ ∈ p [X] and the proof is completed.
= p [X].
(5.7)
SOME PROPERTIES OF BANACH-VALUED SEQUENCE SPACES p [X]
299
Theorem 5.3. A subset B of p [X] (1 < p < ∞) is a relatively σ (p [X], p [X]× )sequentially compact if and only if
(i) B is bounded;
(ii) Pi (B) is a relatively σ (X, X ∗ )-sequentially compact subset of X for each i ∈ N.
Proof. Suppose that B is a relatively σ (p [X], p [X]× )-sequentially compact subset of p [X]. Then B is σ (p [X], p [X]× )-bounded. By Theorem 3.3, B is bounded and
(i) holds. Since Pi is continuous and hence, σ (p [X], p [X]× )-σ (X, X ∗ ) continuous for
each i ∈ N, (ii) holds.
On the other hand, suppose (i) and (ii) hold. Let {x̄ (n) }∞
1 ⊆ B. Using the diagonal
method, by (ii), there exist a subsequence {x̄ (nk ) }k≥1 of {x̄ (n) }n≥1 and xi ∈ X such that
(n )
σ X, X ∗ - lim xi k = xi ,
k
i = 1, 2, . . . .
(5.8)
By (i) and Lemma 5.2, x̄ = (xi )i ∈ p [X]. By Theorem 4.4, σ (p [X], p [X]× )- limk x̄ (nk )
= x̄. So we have proved that B is relatively σ (p [X], p [X]× )-sequentially compact.
This completes the proof.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
P. Cembranos, Some properties of the Banach space c0 (E), Mathematics Today, GauthierVillars, Paris, 1982, pp. 333–336. MR 84a:46045. Zbl 493.46021.
, The hereditary Dunford-Pettis property for 1 (E), Proc. Amer. Math. Soc. 108
(1990), no. 4, 947–950. MR 90i:46019. Zbl 833.46007.
N. K. De Grande-De, Generalized sequence spaces, Bull. Soc. Math. Belg. 23 (1971), 123–
166. MR 46 #9688. Zbl 232.46017.
, Criteria for nuclearity in terms of generalized sequence spaces, Arch. Math. (Basel)
28 (1977), no. 6, 644–651. MR 57 #1070. Zbl 359.46011.
, Operators factoring through a generalized sequence space; applications, Math.
Nachr. 95 (1980), 79–88. MR 84b:46011. Zbl 0453.46008.
M. A. Fugarolas, On Besselian Schauder bases in p (E), Monatsh. Math. 97 (1984), no. 2,
99–105. MR 85h:46024. Zbl 531.46009.
W. Govaerts, Bornological spaces of type c0 (E), Portugal. Math. 41 (1982), no. 1-4, 51–55.
MR 86f:46003. Zbl 545.46026.
M. Gupta, The generalized spaces 1 (X) and m0 (X), J. Math. Anal. Appl. 78 (1980), no. 2,
357–366. MR 82c:46010. Zbl 453.46012.
M. Gupta and Q. Bu, On Banach-valued GAK-sequence spaces p [X], J. Anal. 2 (1994),
103–113. MR 95f:46008. Zbl 818.46023.
M. Gupta, P. K. Kamthan, and J. Patterson, Duals of generalized sequence spaces, J. Math.
Anal. Appl. 82 (1981), no. 1, 152–168. MR 82j:46006. Zbl 492.46010.
P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, New York, 1981.
MR 83g:46011. Zbl 447.46002.
I. E. Leonard, Banach sequence spaces, J. Math. Anal. Appl. 54 (1976), no. 1, 245–265.
MR 54 #8230. Zbl 343.46010.
R. L. Li and Q. Y. Bu, Locally convex spaces containing no copy of c0 , J. Math. Anal. Appl.
172 (1993), no. 1, 205–211. MR 94a:46002. Zbl 779.46012.
A. Marquina and J. M. Sanz-Serna, Barrelledness conditions on c0 (E), Arch. Math. (Basel)
31 (1978/79), no. 6, 589–596. MR 80i:46010. Zbl 396.46005.
A. Marquina and J. Schmets, On bornological c0 (E) spaces, Bull. Soc. Roy. Sci. Liège 51
(1982), no. 5-8, 170–173. MR 84f:46013. Zbl 514.46025.
J. Mendoza, A barrelledness criterion for c0 (E), Arch. Math. (Basel) 40 (1983), no. 2, 156–
158. MR 84k:46006. Zbl 503.46026.
300
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
QINGYING BU
C. Piñeiro, ℵ0 -quasibarrelledness on c0 (E), Arch. Math. (Basel) 53 (1989), no. 1, 61–64.
MR 90f:46008. Zbl 655.46004.
A. Pietsch, Verallgemeinerte Vollkommene Folgenräume, Akademie-Verlag, Berlin, 1962.
MR 28 #452. Zbl 105.30701.
, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math. 28
(1967), 333–353. MR 35 #7162. Zbl 0156.37903.
, Nuclear locally convex spaces, Springer-Verlag, New York, 1972. MR 50#2853.
Zbl 236.46001.
W. H. Ruckle, Sequence Spaces, Pitman (Advanced Publishing Program), Massachusetts,
1981. MR 83g:46012. Zbl 491.46007.
C. Swartz, The Schur lemma for bounded multiplier convergent series, Math. Ann. 263
(1983), no. 3, 283–288. MR 84h:46015. Zbl 506.40006.
A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 1978.
MR 81d:46001. Zbl 395.46001.
C. X. Wu and Q. Y. Bu, Vector-valued sequence space Λ[X] and its Köthe dual. I. GAKproperty, separability and barrelledness, Northeast. Math. J. 8 (1992), no. 3, 275–
282. MR 94g:46010. Zbl 807.46010.
, Characterizations of cmc(X) which is GAK-space, J. Harbin Inst. Tech. 25 (1993),
no. 1, 93–96. MR 94g:46011.
, Köthe dual of Banach sequence spaces p [X] (1 ≤ p < ∞) and Grothendieck
space, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 265–273. MR 95e:46022.
Zbl 785.46009.
Qingying Bu: Department of Mathematical Sciences, Kent State University, Kent,
OH 44242, USA
E-mail address: qbu@math.kent.edu