p-SUMMABLE SEQUENCE SPACES WITH INNER PRODUCTS
HENDRA GUNAWAN1∗ , SUKRAN KONCA2 AND MOCHAMMAD IDRIS3
Abstract. We revisit the space ℓp of p-summable sequences of real numbers.
In particular, we show that this space is actually contained in a (weighted) inner
product space. The relationship between ℓp and the (weighted) inner product
space that contains ℓp is studied. For p > 2, we also obtain a result which
describe how the weighted inner product space is associated to the weights.
1. Introduction
By ℓp = ℓp (R) we denote the space of all p-summable sequences of real numbers.
We know that for p ̸= 2, the space ℓp is not an inner product space, since the
∑
1
p p
p
usual norm ∥x∥p = ( ∞
k=1 |xk | ) on ℓ does not satisfy the paralellogram law.
As an infinite dimensional normed space, ℓp can be equipped with another norm
∑
xk p p1
which is not equivalent to the usual norm. For example, ∥x∥∗ := ( ∞
|
k=1 k | ) , is
such a norm (see [4]). However, one may observe that this norm does not satisfy
the paralellogram law too for p ̸= 2.
One question arises: can we define a norm on ℓp which satisfies the paralellogram law? The reason why we are interested in the paralellogram law is because
we eventually wish to define an inner product, possibly with weights, on ℓp , so
that we can define orthogonality and many other notions on this space. One alternative is to define a semi-inner product on ℓp as in [6], but having a semi-inner
product is not as nice as having an inner product.
The reader might agree that inner product spaces, which were initially introduced by D. Hilbert [3] in 1912, have been up to now the most useful spaces in
practical applications of functional analysis.
In this paper, we introduce a (weighted) inner product on ℓp where 1 ≤ p < ∞.
We discuss the properties of the induced norm and its relationship with the usual
norm on ℓp . We also find that the inner product is actually defined on a larger
space. We study the relationship between ℓp and this larger space, and found
many interesting results, which we shall present in the following sections.
Throughout the paper, we assume that X is a real vector space. As in [5], the
norm on X is a mapping ∥ · ∥ : X → R such that for all vectors x, y ∈ X and
scalars α ∈ R we have:
(1) ∥x∥ ≥ 0 and ∥x∥ = 0 if and only if x = 0,
Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗
Corresponding author.
2000 Mathematics Subject Classification. Primary 46C50; Secondary 46B20; 46C05;
46C15;46B99.
Key words and phrases. Inner product spaces; normed spaces; p-summable sequences;
weights.
1
2
HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS
(2) ∥αx∥ = |α| ∥x∥,
(3) ∥x + y∥ ≤ ∥x∥ + ∥y∥.
The inner product on X is a mapping ⟨·, ·⟩ from X × X into R such that for all
vectors x, y, z ∈ X and scalars α we have:
(1) ⟨x, x⟩ ≥ 0 and ⟨x, x⟩ = 0 if and only if x = 0;
(2) ⟨x, y⟩ = ⟨y, x⟩,
(3) ⟨αx, y⟩ = α ⟨x, y⟩,
(4) ⟨x1 + x2 , y⟩ = ⟨x1 , y⟩ + ⟨x2 , y⟩.
As we work with sequence spaces of real numbers, we will use the sum notation
∞
∑
∑
, for brevity.
instead of
k
k=1
2. Results for 1 ≤ p ≤ 2
In this section, we let 1 ≤ p ≤ 2, unless otherwise stated. First, we observe
that ℓp ⊆ ℓ2 (as sets). Indeed, if x = (xk ) is a sequence in ℓp , then
∑
|xk |2
∥x∥22 =
k
=
∑
|xk |2−p |xk |p
k
≤ sup |xk |2−p
∑
k
≤
[∑
k
|xk |p
k
=
[∑
|xk |p
|xk |p
] 2−p
∑
p
] p2
|xk |p
k
.
k
Taking the square roots of both sides, we get
[∑
] 1 [∑
]1
2 2
p p
|xk |
≤
|xk | ,
k
k
which means that x is in ℓ2 .
Thus we realize that ℓp can actually be considered as a subspace of ℓ2 , equipped
with the inner product
∑
⟨x, y⟩ :=
xk yk ,
(2.1)
k
and the norm
∥x∥2 :=
[∑
|xk |2
] 12
.
(2.2)
k
Being an induced norm from the inner product, the norm ∥ · ∥2 of course satisfies
the paralellogram law:
∥x + y∥22 + ∥x − y∥22 = 2∥x∥22 + 2∥y∥22 ,
for every x, y ∈ ℓp .
p-SUMMABLE SEQUENCE SPACES WITH INNER PRODUCTS
3
A more general result is formulated in the following proposition, which describes the monotonicity property of the norms on ℓp spaces.
Proposition 2.1. If 1 ≤ p ≤ q ≤ ∞, then ℓp ⊆ ℓq , with ∥x∥q ≤ ∥x∥p for every
x ∈ ℓp . Moreover, the inclusion is strict: if 1 ≤ p < q ≤ ∞, then we can find a
sequence x in ℓq which is not in ℓp .
Proof. Let 1 ≤ p ≤ q ≤ inf ty. For every x ∈ ℓp , we have
∑
∥x∥qq =
|xk |q
k
∑
=
|xk |q−p |xk |p
k
≤ sup |xk |q−p
∑
k
[∑
≤
k
|xk |p
k
[∑
=
|xk |p
|xk |p
] q−p
∑
p
] pq
|xk |p
k
.
k
Taking the q-th roots of both sides, we get
∥x∥q ≤ ∥x∥p ,
which tells us that ℓ ⊆ ℓ .
( 1 )
To show that the inclusion is strict, one may take x = (xk ) := k1/p
, where
q
p
1 ≤ p < ∞. Clearly x ∈ ℓ for p < q ≤ ∞, but x ∈
/ℓ.
p
q
When we equip ℓp , where 1 ≤ p < 2, with ∥ · ∥2 , one might ask whether ∥ · ∥2 is
equivalent with the usual norm ∥ · ∥p . The answer is negative. We already have
∥x∥2 ≤ ∥x∥p for every x ∈ ℓp . The following proposition tells us that we cannot
control ∥x∥p by ∥x∥2 for every x ∈ ℓp .
Proposition 2.2. Let 1 ≤ p < 2. There is no constant C > 0 such that ∥x∥2 ≥
C ∥x∥p for every x ∈ ℓp .
(
)
Proof. For each n ∈ N, take x(n) := 11+ 1 . Then,
∥x(n) ∥22 =
kp
∑
1
k
k p+n
2
while
∥x(n) ∥pp =
2
∑
≤
∑ 1
2
k
1
kp
n
< ∞ (independent of n),
< ∞ (dependent of n),
k
k
which tends to ∞ as n → ∞. Hence
∥x(n) ∥2
→ 0, as n → ∞.
∥x(n) ∥p
p
1+ n
4
HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS
So, there is no constant C > 0 such that ∥x∥2 ≥ C∥x∥p for every x ∈ ℓp .
Proposition 2.3. Let 1 ≤ p < 2. As a set in (ℓ2 , ∥ · ∥2 ), ℓp is not closed but
dense in ℓ2 .
Proof.
To show that ℓp )is not closed in ℓ2 , for each n ∈ (N we
take x(n) :=
(
)
1
1
1
1, 21/p
, . . . , n1/p
, 0, 0, . . . . Clearly (x(n) ) converges to x := k1/p
in ∥ · ∥2 . But
(n)
p
p
p
x ∈ ℓ for each n ∈ N, while x ∈
/ ℓ . Therefore ℓ is not closed in ℓp .
p
2
To show that ℓ is dense in ℓ , we observe that every x = (xk ) ∈ ℓ2 can be
approximated arbitrarily close by x(n) := (x1 , . . . , xn , 0, 0, . . . ) ∈ ℓp , n ∈ N.
Remark 2.4. Propositions 2.2 and 2.3 warn us that when we use the ℓ2 -inner
product and its induced norm on ℓp for 1 ≤ p < 2, we have to be careful –
especially when we deal with the topology.
3. Results for 2 < p < ∞
Throughout this section, we let 2 < p < ∞, unless otherwise stated. As we
have seen in Proposition 2.1, the space ℓp is now larger than ℓ2 . Thus the usual
inner product and norm on ℓ2 are not defined for all sequences in ℓp .
To define an inner product and a new norm on ℓp which satisfies the paralellogram law, we have to use weights. Let us choose v = (vk ) ∈ ℓp where
vk > 0, k ∈ N. Next we define the mapping ⟨·, ·⟩v which maps every pair of
sequences x = (xk ) and y = (yk ) to
∑ p−2
vk xk yk ,
(3.1)
⟨x, y⟩v :=
k
and the mapping ∥ · ∥2,v which maps every sequence x = (xk ) to
[∑
] 12
∥x∥2,v :=
vkp−2 |xk |2 .
(3.2)
k
We observe that the mappings are well defined on ℓp . Indeed, for x = (xk ) and
y = (yk ) in ℓp , it follows from Hölder’s inequality that
[∑ ] p−2
[∑
] p1 [∑
] p1
∑ p−2
p
vk xk yk ≤
vkp
|xk |p
|yk |p .
(3.3)
k
k
k
k
p
Thus the two mappings are defined on ℓ . Moreover, we have the following
proposition, whose proof is left to the reader.
Proposition 3.1. The mappings in (3.1) and (3.2) define a weighted inner product and a weighted norm, respectively, on ℓp .
From (3.3), we see that the following inequality
p−2
∥x∥2,v ≤ ∥v∥p 2 ∥x∥p
holds for every x ∈ ℓp . It is then tempting to ask whether the two norms are
equivalent on ℓp . The answer, however, is negative, due to the following result.
p-SUMMABLE SEQUENCE SPACES WITH INNER PRODUCTS
5
Proposition 3.2. There is no constant C = Cv > 0 such that
∥x∥p ≤ C ∥x∥2,v ,
for every x ∈ ℓp .
Proof. Suppose that such a constant exists. Then, for x := en = (0, . . . , 0, 1, 0, . . . ),
where the 1 is the nth term, we have
p−2
1 ≤ C vn 2 .
But this cannot be true, since vn → 0 as n → ∞.
According to Proposition 3.2, it is possible for us to find a sequence in ℓp which
is divergent with respect to the norm ∥ · ∥p , but convergent with respect to the
norm ∥ · ∥2,v .
Example 3.3. Let x(n) := en ∈ ℓp , where en = (0, . . . , 0, 1, 0, . . . ) (the 1 is the
1
nth term), then ∥x(m) − x(n) ∥p = 2 p 9 0 as m, n → ∞. Since (x(n) ) is not a
Cauchy sequence with respect to ∥ · ∥p , it is not convergent with respect to ∥ · ∥p .
p−2
However, ∥x(n) ∥2,v = vn 2 → 0 as n → ∞, since vn → 0 as n → ∞. Hence, (x(n) )
is convergent with respect to the norm ∥ · ∥2,v .
If we wish, we can also define another weighted norm ∥ · ∥β,v on ℓp , where
1 ≤ β ≤ p < ∞, by
[∑
]1
p−β
β β
∥x∥β,v :=
vk |xk |
.
k
Here p may be less than 2. Note that if β = p, then ∥ · ∥β,v = ∥ · ∥p .
The following proposition gives a relationship between two such weighted norms
on ℓp .
p(γ−β)
γ
Proposition 3.4. Let 1 ≤ β < γ ≤ p. Then we have ∥x∥β,v ≤ ∥v∥p
for every x ∈ ℓp .
∥x∥γ,v
Proof. Suppose that x ∈ ℓp . We compute
]1
[∑
] 1 [∑ p(γ−β) (p−γ)β
p−β
γ
γ
β β
β β
vk
|xk |
∥x∥β,v =
vk |xk |
=
vk
k
≤
[∑
vkp
[∑
] γ−β
γβ
vkp−γ |xk |γ
] γ1
k
k
p(γ−β)
γβ
= ∥v∥p
as desired.
k
∥x∥γ,v ,
6
HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS
Corollary 3.5. If 1 ≤ β < 2 < γ ≤ p, then there are constants C1,v , C2,v > 0
such that
C1,v ∥x∥β,v ≤ ∥x∥2,v ≤ C2,v ∥x∥γ,v
p
for every x ∈ ℓ .
4. Further Results
Let 2 < p < ∞. As we have seen in the previous section, every sequence x ∈ ℓp
has ∥x∥2,v < ∞. This suggests that ℓp lives inside a larger space, consisting all
sequences x with ∥x∥2,v < ∞. Let us denote this space by ℓ2v . We shall now
discuss some properties of this space. First, we have the following proposition,
which describes the relationship between ℓ2v and ℓp .
Proposition 4.1. As sets, we have ℓp ⊂ ℓ2v and the inclusion is strict.
Proof. Let x ∈ ℓp . It follows from (3.3) that
p−2
∥x∥2,v ≤ ∥v∥p 2 ∥x∥p ,
which means that x ∈ ℓ2v .
show that the inclusion
is strict, we need to find x = (xk ) such that
∑
∑To p−2
p
2
k |xk | = ∞. We know that vk > 0 for all k ∈ N,
k vk |xk | < ∞ but
p−2
and vk → 0 as k → ∞. So, choose m1 ∈ N such that vm
< 12 , m2 > m1 such
1
p−2
p−2
that vm
< 213 , and so on. Since the process never
< 212 , m3 > m2 such that vm
3
2
stops, we obtain an increasing sequence of nonnegative integers (mj ) such that
p−2
vm
< 2−j for every j ∈ N. Now put xk := 1 for k = m1 , m2 , m3 , . . . and xk := 0
j
otherwise. Hence
∑ 1
∑
∑ p−2
p−2
<
vk |xk |2 =
vm
= 1,
j
j
2
j
j
k
∑
∑
p
while k |xk | = j 1 = ∞. This means that x is in ℓ2v but not in ℓp .
Theorem 4.2. The space (ℓ2v , ∥ · ∥2,v ) is complete. Accordingly, (ℓ2v , ⟨·, ·⟩v ) is a
Hilbert space.
Proof. It is easy to see that the space (ℓ2v , ∥ · ∥2,v ) is a linear normed space, so we
omit the details. To prove the completeness, let (x(n) ) be any Cauchy sequence
(n)
(n)
(n)
in the space ℓ2v , where x(n) = (xk ) = (x1 , x2 , . . . ). Then for every ε > 0 there
is an n0 ∈ N such that for all n, m > n0
] 12
[∑
(n)
(m)
∥x(n) − x(m) ∥2,v =
vkp−2 |xk − xk |2 < ε.
(4.1)
k
It follows that for each k ∈ N we have
∑ p−2 (n)
(n)
(m)
(m)
vkp−2 |xk − xk |2 ≤
vk |xk − xk |2 = ∥x(n) − x(m) ∥22,v < ε2 .
k
Then
(n)
(m)
2−p
|xk − xk | ≤ εvk 2
p-SUMMABLE SEQUENCE SPACES WITH INNER PRODUCTS
7
(n)
for all m, n > n0 . Thus, for each fixed k ∈ N, (xk ) is a Cauchy sequence of real
(n)
numbers. Hence, it is convergent, say xk → xk as n → ∞.
Using these infinitely many limits x1 , x2 , x3 , . . . , we define the sequence x :=
(x1 , x2 , x3 , . . . ). Then we have constructed a candidate limit for the sequence
(x(n) ). However, so far, we only have that each individual component of x(n) converges to the corresponding component of x, i.e., (x(n) ) converges componentwise
to x. To prove that (x(n) ) converges to x in norm, we go back to (4.1) and pass
it to the limit m → ∞. We obtain
]1
[∑
p−2 (n)
2 2
(n)
∥x − x∥2,v =
vk |xk − xk |
< ε,
k
for all n > n0 . Since the space ℓ2v is linear, we also get x = (x − xn ) + xn ∈ ℓ2v .
This completes the proof.
The following proposition tells us that ℓp is “not too far” from ℓ2v .
Proposition 4.3. As a set in (ℓ2v , ∥ · ∥2,v ), ℓp is not closed but dense in ℓ2v .
Proof. As in the proof of Proposition 4.1, we construct an increasing sequence of
p−2
nonnegative integers (mj ) such that vm
< 2−j for every j ∈ N. Next, for each
j
(n)
(n)
(n)
j ∈ N, we define x(n) = (xk ) by xk := 1 for k = m1 , m2 , . . . , mn and xk := 0
otherwise. Then we see that (x(n) ) forms a Cauchy sequence in ℓ2v since for m > n
we have
m
∑ 1
∑
∑
(n)
(m)
p−2
<
vkp−2 |xk − xk |2 ≤
vm
∥x(n) − x(m) ∥22,v =
→ 0,
j
j
2
m >n
j>n
k=n+1
j
as n → ∞. Since ℓ2v is complete, (x(n) ) is convergent and we know that the
limit is the sequence x = (xk ) where xk = 1 for k = m1 , m2 , m3 , . . . and xk = 0
otherwise. While x(n) ∈ ℓp for every n ∈ N, we see that the limit x ∈
/ ℓp . This
p
2
shows that ℓ is not closed in ℓv .
The fact that ℓp is dense in ℓ2v is easy to see, since every x = (xk ) ∈ ℓ2v can be
approximated by x(n) := (x1 , . . . , xn , 0, 0, . . . ) for sufficiently large n ∈ N.
Proposition 4.3 motivates us to study ℓ2v further as the ambient space, replacing
ℓp . So far, we have fixed the weight v = (vk ). We now would like to know how
the space ℓ2v depends on the choice of v.
Let Vp be the collection of all sequences v = (vk ) ∈ ℓp with vk > 0 for every
k ∈ N. Let v = (vk ), w = (wk ) ∈ Vp . We say that v and w are equivalent and
write v ∼ w if and only if there exists a constant C > 0 such that
1
vk ≤ w k ≤ C v k
C
for every k ∈ N. Our final theorem is the following.
8
HENDRA GUNAWAN, SUKRAN KONCA, MOCHAMMAD IDRIS
Theorem 4.4. Let v, w ∈ Vp . Then, the following statements are equivalent:
(1) v ∼ w.
(2) There exists a constant C > 0 such that
∥x∥2,w ≤ C ∥x∥2,v ,
x ∈ ℓ2v ,
and
∥x∥2,v ≤ C ∥x∥2,w , x ∈ ℓ2w .
(3) ℓ2v = ℓ2w as sets, and the two norms ∥ · ∥2,v and ∥ · ∥2,w are equivalent there.
(4) There exists a constant C > 0 such that
1
∥x∥2,v ≤ ∥x∥2,w ≤ C ∥x∥2,v , x ∈ ℓp .
C
Proof. The chain of implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are clear. Hence it
remains only to show that (4) ⇒ (1). Assume that there exists a constant C > 0
such that
1
∥x∥2,v ≤ ∥x∥2,w ≤ C ∥x∥2,v , x ∈ ℓp .
C
Take x := en , where n ∈ N is fixed but arbitrary. Then x ∈ ℓp , so that x is in ℓ2v
as well as in ℓ2w . Moreover,
p−2
∥x∥2,v = vn 2
p−2
and ∥x∥2,w = wn 2 .
Hence, from our assumption, we obtain
p−2
p−2
1 p−2
vn 2 ≤ wn 2 ≤ Cvn 2 ,
C
( )
and this holds for every n ∈ N. Taking the p−2
-th roots, we conclude that
2
v ∼ w. This completes the proof.
Remark 4.5. Theorem 4.4 is interesting in the sense that the two spaces (ℓ2v , ∥·∥2,v )
and (ℓ2w , ∥ · ∥2,w ) are identical if and only if ∥ · ∥2,v and ∥ · ∥2,w are equivalent on ℓp ,
and this can be checked through
(
)
( 1 ) the equivalence of v and w. Thus, for example,
the space associated to u := k is identical with that associated to v := k2k+1 ,
( )
but is different from the one associated to w := k12 . We also note that, according
to Proposition 4.3, even though ℓ2v and ℓ2w may be different, both spaces contain
ℓp as a dense subset.
5. Concluding Remarks
We have shown that the space ℓp can be equipped with a (weighted) inner product and its induced norm. Using the inner product, one may define orthogonality
on ℓp , carry out the Gram-Schmidt process to get an orthogonal set, define the
volume of an n-dimensional parallelepiped on ℓp (see [2]), and so on. When we
have to deal with topology, however, we are suggested to consider the (weighted)
inner product space that contains ℓp , namely ℓ2 for 1 ≤ p ≤ 2 or ℓ2v for 2 < p < ∞
(where v ∈ ℓp with vk > 0 for every k ∈ N), which are complete.
p-SUMMABLE SEQUENCE SPACES WITH INNER PRODUCTS
9
There we might also be interested in bounded linear functionals. For example,
for 2 < p < ∞, the functional fy := ⟨·, y⟩v is linear and bounded on ℓ2v , and its
norm can be given by
|fy (x)|
|⟨x, y⟩v |
∥fy ∥ = sup
= sup
.
∥x∥2,v ̸=0 ∥x∥2,v
∥x∥2,v ̸=0 ∥x∥2,v
Clearly ∥fy ∥ ≤ ∥y∥2,v , and by taking x := ∥y∥y2,v we obtain ∥fy ∥ = ∥y∥2,v . Moreover, we can prove an analog of the Riesz-Fréchet Theorem (see [1]), which states
that for any bounded linear functional g on ℓ2v , there exists a unique y ∈ ℓ2v such
that g(x) = ⟨x, y⟩v for every x ∈ ℓ2v and ∥g∥ = ∥y∥2,v .
All the results for 2 < p < ∞ also hold for Lp (R), the space of p-integrable
functions on R. For 1 ≤ p < 2, however, different situations take place.
References
[1] Berberian, S.K. (1961): Introduction to Hilbert Space. New York: Oxford Univ. Press.
[2] Gunawan, H.; W. Setya-Budhi; Mashadi; S. Gemawati (2005): On volumes of ndimensional parallelepipeds on ℓp spaces, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.
16, 48–54.
[3] Hilbert, D. (1912): Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen.
Repr. 1953, New York: Chelsea.
[4] Idris, M.; S. Ekariani; H. Gunawan (2013): On the space of p-summable sequences. Mat.
Vesnik. 65:1, 58–63.
[5] Kreyszig, E. (1978): Introductory Functional Analysis with Applications. New York: John
Wiley & Sons.
[6] Miličić, P.M. (1987): Une généralisation naturelle du produit scaleaire dans un espace
normé et son utilisation. Publ. Inst. Math. (Beograd) 42:56, 63–70.
1
Department of Mathematics, Institut Teknologi Bandung, Bandung 40132,
Indonesia.
E-mail address: 1 [email protected]
2
Department of Mathematics, Sakarya University, 54187, Sakarya, Turkey.
E-mail address: 2 [email protected]
3
Department of Mathematics, Institut Teknologi Bandung, Bandung 40132,
Indonesia.
E-mail address: 3 [email protected]