MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 1 (2013), 58–63
March 2013
ON THE SPACE OF p-SUMMABLE SEQUENCES
M. Idris, S. Ekariani and H. Gunawan
Abstract. On the space `p of p-summable sequences (of real numbers), one can derive a
norm from the 2-norm as indicated by Gunawan [H. Gunawan, The space of p-summable sequences
and its natural n-norms, Bull. Austral. Math. Soc. 64 (2001), 137–147]. The purpose of this
note is to establish the equivalence between such a norm and the usual norm on `p . We show that
our result is useful in understanding the topology of `p as a 2-normed space.
1. Introduction
In 2001, Gunawan [9] showed that the space `p = `pR (N) of p-summable sequences (of real numbers) can be equipped with a 2-norm. In general, if X is a
(real) vector space, then a mapping k·, ·k : X × X → R which satisfies the following
properties:
(1.1) kx, yk = 0 if and only if x and y are linearly dependent,
(1.2) kx, yk = ky, xk for every x, y ∈ X,
(1.3) kαx, yk = |α| kx, yk for every x, y ∈ X and α ∈ R,
(1.4) kx + y, zk ≤ kx, zk + ky, zk for every x, y, z ∈ X,
is called a 2-norm on X, and the pair (X, k·, ·k) is called a 2-normed space. The
concept of 2-normed spaces was first introduced by Gähler [3] and its generalization
can be found in [5–7]. Related works may be found in [2, 4, 8, 12, 14]. See also [1,
10, 11, 13] for some results on this subject in the last decade.
For example, on the space `p (1 ≤ p ≤ ∞) that we are interested in now, the
following mapping
¯
¯ ¸1/p
·
1 P P¯¯ ¯¯ xk xl ¯¯ ¯¯p
kx, ykp :=
,
(1)
¯¯
¯
2 k l yk yl ¯
where x = (xk ) and y = (yk ), defines a 2-norm. For p = ∞, the formula reduces to
¯¯
¯ ¯¯
¯ x xl ¯¯ ¯
kx, yk∞ := sup sup¯ ¯¯ k
¯.
yk yl ¯
k
l
2010 AMS Subject Classification: 46B05, 46B20, 46A45, 46B45, 46B99.
Keywords and phrases: p-summable sequence spaces; norm equivalence.
58
On the space of p-summable sequences
We also note that, for p = 2, the 2-norm may be rewritten as
¯
¯
¯ hx, xi hx, yi ¯1/2
¯
¯ ,
kx, yk2 = ¯
hy, xi hy, yi ¯
59
(2)
where hx, yi denotes the usual inner product on `2 . (Geometrically, in an inner
product space (X, h·, ·i), the entity kx, yk2 given by (2) represents the area of the
parallelogram spanned by x and y in X.)
Now, as shown in [9], we can derive a norm from the 2-norm in a certain way.
Indeed, if e1 := (1, 0, 0, . . . ) and e2 := (0, 1, 0, . . . ), then the following function
£
¤1/p
kxk := kx, e1 kpp + kx, e2 kpp
(3)
defines a norm on `p . In general, if {a, b} is a linearly independent set in `p , then
one may observe that
£
¤1/p
kxk◦ := kx, akpp + kx, bkpp
(4)
defines a norm on `p . If the former can be easily seen to be equivalent to the usual
norm k · kp on `p , the latter is not so obvious.
The aim of this note is to prove the equivalence between the usual norm k · kp
and the norm k · k◦ defined in (4), for any linearly independent set {a, b} in `p . This
result is important because in an infinite dimensional space there is no guarantee
that any two given norms are equivalent. As we shall see, our result can also be
used to understand the topology of `p as a 2-normed space.
2. Main results
Hereafter we shall consider only the case where 1 ≤ p < ∞, leaving the case
where p = ∞ to the reader. We assume that the reader has sufficient knowledge of
`p space, especially its completeness.
Before we present our results, we shall first show by example that on `p there
are norms which are not equivalent to the usual one. For every x = (xk ) in `p ,
define
hP¯ x ¯p i1/p
¯ k¯
kxk∗ :=
.
¯ ¯
k k
Clearly k · k∗ defines a norm on `p . Here we have kxk∗ ≤ kxkp for every x ∈ `p .
However, we cannot find a constant A > 0 such that kxk∗ ≥ Akxkp for every x ∈ `p .
If there were such a constant A > 0, we could take x = ek := (0, · · · , 0, 1, 0, · · · )
where the only nonzero term is the k-th term, so that k1 ≥ A. But this cannot be
true for any k ∈ N, for otherwise N would be bounded by A1 .
Let us now examine the norm k · k derived from the 2-norm k·, ·kp through the
formula (3). For every x ∈ `p , the formula (1) gives
kx, e1 kpp = |x2 |p + |x3 |p + |x4 |p + · · · ,
kx, e2 kpp = |x1 |p + |x3 |p + |x4 |p + · · · ,
60
M. Idris, S. Ekariani, H. Gunawan
whence
kxkp = |x1 |p + |x2 |p + 2(|x3 |p + |x4 |p + · · · ).
Thus we have kxkp ≤ kxk ≤ 21/p kxkp , for every x ∈ `p . This tells us that k · k is
equivalent to the usual norm k · kp .
The norm k · k defined by (3) is just a special case of the norm k · k◦ defined by
(4) for any linearly independent set {a, b} in `p . To prove that k · k◦ is equivalent to
the usual norm k · kp , we need the following lemmas. (Lemma 2.1 is taken from [9].)
Lemma 2.1. For every x, y ∈ `p , we have kx, ykp ≤ 21−1/p kxkp kykp .
Proof. By the triangle inequality for real numbers and Minkowski’s inequality
for double series, we have
¸1/p
·
1 PP
p
kx, ykp =
|xk yl − xl yk |
2 k l
·
¸1/p
1 PP
p
≤
(|xk ||yl | + |xl ||yk |)
2 k l
·h
i1/p hP P
i1/p ¸
PP
−1/p
p
p
p
p
≤2
|xk k |yl |
+
|xl | |yk |
k
l
k
l
= 21−1/p kxkp kykp ,
for every x, y ∈ `p , as claimed.
Lemma 2.2. For every x, y, z ∈ `p , we have kxkp ky, zkp ≤ 2 kykp kx, zkp +
kzkp kx, ykp .
p
Proof. For every x, y, z ∈ `p , we have ky, zkp =
P
p
p
kxkp = k |xk | , so that
p
p
kxkp ky, zkp =
1
2
P P
m
p
n
|ym zn − zm yn | and
1 PPP
p
|xk ym zn − xk yn zm | .
2 k m n
Now observe that
1 PPP
p
|xk ym zn − xk yn zm |
2 k m n
1 PPP
p
=
|(xk ym zn − xn ym zk ) + (xn ym zk − xm yn zk ) + (xm yn zk − xk yn zm )|
2 k m n
∞ P
∞ P
∞ ¡
¢p
1 P
≤
|xk ym zn − xn ym zk | + |xn ym zk − xm yn zk | + |xm yn zk − xk yn zm |
2 k=1 m=1 n=1
It follows from Minkowski’s inequality that
³1 P P P
2
k m n
p
|xk ym zn − xk yn zm |
´ p1
≤
³1 P P P
2
k m n
p
|xk ym zn − xn ym zk |
´ p1
+
61
On the space of p-summable sequences
+
³1 P P P
|xn ym zk − xm yn zk |
p
´ p1
+
³1 P P P
2 k m n
2
= kykp kx, zkp + kzkp kx, ykp + kykp kx, zkp
k m n
|xm yn zk − xk yn zm |
p
´ p1
Thus we obtain
kxkp ky, zkp ≤ 2 kykp kx, zkp + kzkp kx, ykp ,
as desired.
Now we come to the main result.
Theorem 2.3. For any linearly independent set {a, b} in `p , the norm k · k◦
defined by (4) is equivalent to the usual norm k · kp .
1
1
Proof. For every x ∈ `p , we have kx, akp ≤ 21− p kxkp kakp and kx, bkp ≤
21− p kxkp kbkp by Lemma 2.1. Hence we obtain
h
i1
h
i1
p
p p
p
p p
kxk◦ = kx, akp + kx, bkp ≤ 21−1/p kakp + kbkp kxkp .
Meanwhile, by Lemma 2.2, we have kxkp ka, bkp ≤ 2 kakp kx, bkp + kbkp kx, akp and
by swapping a and b we get kxkp kb, akp ≤ 2 kbkp kx, akp + kakp kx, bkp . Hence
2 kxkp ka, bkp ≤ 3 kakp kx, bkp + 3 kbkp kx, akp .
h
i1
p
p p
Next, we know that kx, akp ≤ kx, akp + kx, bkp
= kxk◦ and similarly kx, bkp
≤ kxk◦ . It thus follows that
2 kxkp ka, bkp ≤ 3 (kakp + kbkp )kxk◦ ,
whence
kxkp ≤
3(kakp + kbkp )
kxk◦ .
2ka, bkp
Combining this and the previous inequality, we obtain
h
i1/p
2ka, bkp
kxkp ≤ kxk◦ ≤ 21−1/p kakpp + kbkpp
kxkp .
3(kakp + kbkp )
This shows that k · k◦ and k · kp are equivalent.
Corollary 2.4. The space (`p , k · k◦ ) is complete. In other words, it is a
Banach space.
62
M. Idris, S. Ekariani, H. Gunawan
3. Applications
We recall that a sequence (x(n)) in a 2-normed space (X, k·, ·k) is said to
converge to an element x ∈ X (with respect to k·, ·k) if for every y ∈ X, we have
kx(n) − x, yk → 0, as n → ∞.
Also, a sequence (x(n)) in a 2-normed space (X, k·, ·k) is called a Cauchy sequence
(with respect to k·, ·k) if for every y ∈ X, we have
kx(m) − x(n), yk → 0, as m, n → ∞.
Clearly if (x(n)) converges to an element in (X, k·, ·k), then it is a Cauchy sequence.
If the converse is also true for every sequence in (X, k·, ·k), then X is said to be
complete. A complete 2-normed space is called a 2-Banach space.
By Lemma 2.1, we see that, in `p , if a sequence (x(n)) converges to x with
respect to the usual norm k · kp , then it also converges to x with respect to the
2-norm k·, ·kp . Similarly, if (x(n)) is a Cauchy sequence in `p with respect to k · kp ,
then it is also a Cauchy sequence with respect to k·, ·kp . As a consequence of
Theorem 2.3, we have the following results.
Theorem 3.1. In `p , if a sequence (x(n)) converges to x with respect to k·, ·kp ,
then it also converges to x with respect to k·kp . Also, if (x(n)) is a Cauchy sequence
with respect to k·, ·kp , then it is a Cauchy sequence with respect to k · kp .
Proof. Let {a, b} be a linearly independent set in `p , and k · k◦ be defined by
(4). Now, if (x(n)) converges to x with respect to k·, ·kp , then we have
kx(n) − x, ak → 0, as n → ∞
and
kx(n) − x, bk → 0, as n → ∞.
It follows that
kx(n) − xk◦ → 0, as n → ∞,
that is, (x(n)) converges to x with respect to k · k◦ . By Theorem 2.3, we conclude
that (x(n)) also converges to x with respect to k·kp . The second part of the theorem
is proved in a similar way.
Corolalry 3.2. (`p , k·, ·kp ) is a 2-Banach space.
Proof. Let (x(n)) be a Cauchy sequence in `p with respect to k·, ·kp . Then,
by Theorem 3.1, (x(n)) is a Cauchy sequence with respect to k · kp . We know that
(`p , k · kp ) is a Banach space, and so (x(n)) must converge to an element x ∈ `p
with respect to k · kp . By Lemma 2.1, (x(n)) must also converge to x with respect
to k·, ·kp . Therefore, (`p , k·, ·kp ) is a 2-Banach space.
Our result can also be applied to prove a fixed point theorem in (`p , k·, ·kp ),
which is a refinement of the one in [9].
Theorem 3.3. (Fixed Point Theorem) Let {a, b} be a linearly independent
set in `p , and T be a self-mapping of `p . If there exists a constant K ∈ (0, 1) such
that, for z ∈ {a, b}, we have
kT x − T y, zk ≤ Kkx − y, zk, for every x, y ∈ `p ,
On the space of p-summable sequences
63
then T has a unique fixed point (that is, there exists an element x ∈ `p such that
T x = x).
Remark. In [9], the inequality is assumed to hold for every z ∈ `p . Here we
only require that it holds for z = a and z = b only.
Proof. For every x, y ∈ `p , we observe that
£
¤1/p
kT x − T yk◦ = kT x − T y, akpp + kT x − T y, bkpp
£
¤1/p
≤ K kx − y, akpp + kx − y, bkpp
= K kx − yk◦ .
This tells us that T is a contractive mapping on (`p , k · k◦ ), which is a Banach space
(by Corollary 2.4). Thus T must have a unique fixed point in `p .
Acknowledgements. The second and third authors are supported by ITB
I-MHERE PDU Research Grant 2011. We thank the referees for their useful comments on the earlier version of this paper.
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(received 04.04.2011; in revised form 02.08.2011; available online 10.09.2011)
Department of Mathematics, Universitas Lambung Mangkurat, Banjarbaru, Indonesia
E-mail: awmius@yahoo.com
Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia
E-mail: shelvi ekariani@students.itb.ac.id, hgunawan@math.itb.ac.id