Journal of Inequalities in Pure and
Applied Mathematics
A GENERALIZATION OF HÖLDER AND MINKOWSKI INEQUALITIES
YILMAZ YILMAZ, M. KEMAL ÖZDEMİR AND İHSAN SOLAK
Department of Mathmematics
İnönü University
44280 Malatya / TURKEY
volume 7, issue 5, article 193,
2006.
Received 29 June, 2006;
accepted 09 November, 2006.
Communicated by: K. Nikodem
EMail: yyilmaz@inonu.edu.tr
EMail: kozdemir@inonu.edu.tr
EMail: isolak@inonu.edu.tr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
177-06
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Abstract
In this work, we give a generalization of Hölder and Minkowski inequalities to
normal sequence algebras with absolutely monotone seminorm. Our main result is Theorem 2.1 and Theorem 2.2 which state these extensions. Taking
F = `1 and k·kF = k·k1 in both these theorems, we obtain classical versions of
these inequalities. Also, using these generalizations we construct the vectorvalued sequence space F (X, λ, p) as a paranormed space which is a most
general form of the space c0 (X, λ, p) investigated in [6].
2000 Mathematics Subject Classification: Primary 26D15, 47A30; Secondary
46A45.
Key words: Inequalities, Hölder, Minkowski, Sequence algebra, Vector-valued sequence space.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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1.
Introduction
Hölder and Minkowski inequalities have been used in several areas of mathematics, especially in functional analysis. These inequalities have been generalized in various directions. The purpose of this paper is to give some extensions
of the classical Hölder and Minkowski inequalities. We discovered that the
classical versions are only a type of these extensions in `1 which is a normal
sequence algebra with absolutely monotone seminorm k·k1 .
We now recall some definitions and facts.
A Frechet space is a complete total paranormed space. If H is an Hausdorff
space then an FH-space is a vector subspace X of H which is a Frechet space
and is continuously embedded in H, that is, the topology of X is larger than the
relative topology of H. Moreover if X is a normed FH-space then it is called
a BH-space. An FH-space with H = w, the space of all complex sequences,
is called an FK-space, so a BK-space is a normed FK-space. We know that
`∞ , c, c0 and `p (1 ≤ p < ∞) are BK-spaces. The following relation exists
among these sequence spaces:
`p ⊂ c0 ⊂ c ⊂ `∞ .
A basis for a topological vector spaceP
X is a sequence (bn ) such that every
x ∈ X has
tn bn . This is equivalent to the fact
Pma unique representation x =
that x − n=1 tn bn → 0 (m → ∞) in the vector topology of X. For example,
c0 and `p have (en ) as a basis (en is a sequence x where xn = 1, xk = 0 for
n 6= P
k). If X has a basis (bn ) the functionals ln , given by ln (x) = tn when
x=
tn bn , are linear. They are called the coordinate functionals and (bn ) is
called a Schauder basis if each ln ∈ X 0 , the continuous dual of X. A basis of
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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a Frechet space must be a Schauder basis [7]. An FK-space X is said to have
AK, or be an AK-space, if X ⊃ φ (the space of all finite sequences) and (en )
[n]
[n]
is
P x, where x , the nth section of x is
Pan basis for X, i.e. for each x, x →
xk ek for all x ∈ X [8]. The spaces c0
k=1 xk ek ; otherwise expressed, x =
and `p are AK-spaces but c and `∞ are not. We say that a sequence space F is
an AK-BK space if it is both a BK and an AK-space.
An algebra A over a field K is a vector space A over K such that for each
ordered pair of elements x, y ∈ A a unique product xy ∈ A is defined with the
properties
(1) (xy)z = x(yz)
(2a) x(y + z) = xy + xz
(2b) (x + y)z = xz + yz
(3) α(xy) = (αx)y = x(αy)
for all x, y, z ∈ A and scalars α [4].
If K = R (real field) or C (complex field) then A is said to be a real or
complex algebra, respectively.
Let F be a sequence space and x, y be arbitrary members of F . F is called
a sequence algebra if it is closed under the multiplication defined by xy =
(xi yi ), i ≥ 1, and is called normal or solid if y ∈ F whenever |yi | ≤ |xi |, for
some x ∈ F . If F is both a normal and sequence algebra then it is called a
normal sequence algebra. For example, c is a sequence algebra but not normal.
w, `∞ , c0 and `p (0 < p < ∞) are normal sequence algebras.
A paranorm p on a normal sequence space F is said to be absolutely monotone if p(x) ≤ p(y) for x, y ∈ F with |xi | ≤ |yi | for each i [3].
The norm kxk∞ = sup |xk | which makes the spaces `∞ , c, c0 a BK-space, is
P
p 1/p
absolutely monotone. For p ≥ 1, the norm kxk = ( ∞
over `p is
k=1 |xk | )
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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P
p
absolutely monotone. Also, for 0 < p < 1, the p-norm kxkp = ∞
k=1 |xk | over
`p is absolutely monotone.
An Orlicz function is a function M : [0, ∞) −→ [0, ∞) which is continuous,
non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) →
∞ as x → ∞. We say that the Orlicz function M satisfies the ∆´-condition
if there exist positive constants a and u such that M (xy) ≤ aM (x) M (y)
(x, y ≥ u). By means of M , Lindenstrauss and Tzafriri [2] constructed the
sequence space
X |xk | `M = x ∈ w :
M
< ∞ for some ρ > 0
ρ
n
o
P |xk | with the norm kxkM = inf ρ > 0 :
M ρ ≤ 1 . This norm is absolutely monotone and `M is normal since M is non-decreasing. Also if M satisfies the ∆´-condition then `M is a sequence algebra.
Now we give a useful inequality from classical analysis.
Lemma 1.1. Let f be a function such that f 00 (x) ≥ 0 for x > 0. Then for
0<a<x<b
Z x
f (x) − f (a)
1
=
f 0 (t)dt
x−a
x−a a
≤ f 0 (x)
Z b
1
f 0 (t)dt
≤
b−x x
f (b) − f (x)
=
.
b−x
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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Hence
f (x) ≤
b−x
x−a
f (a) +
f (b)
b−a
b−a
[7].
Apply this to the function f (x) = − ln x with θ = (b − x)/(b − a). Then for
all a, b positive numbers and 0 ≤ θ ≤ 1, we have
(1.1)
aθ b1−θ ≤ aθ + (1 − θ)b.
Next, we give a lemma associated with the theorems in Section 2.
Lemma 1.2.
a) Let F be a normal sequence algebra, u = (un ) ∈ F and p ≥ 1. Then
up = (upn ) ∈ F .
b) If F is a normal sequence space, k·kF is an absolutely monotone seminorm
on F and u = (un ) ∈ F then |u| = (|un |) ∈ F and k|u|kF = kukF .
Proof. a) We define two sequences a = (an ) and b = (bn ) such that
(
(
un if |un | ≥ 1
0 if |un | ≥ 1
an =
and bn =
.
0 if |un | < 1
un if |un | < 1
So un = an + bn and upn = apn + bpn . Obviously, a, b ∈ F . Since p < [p] + 1, we
have
|an |p ≤ |an |[p]+1 ,
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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where [p] denotes the integer part of p. Since F is a sequence algebra, the
sequence a[p]+1 is a member of F by induction, and so ap ∈ F . Furthermore,
since F is normal and |bn |p ≤ |bn |, we have bp ∈ F . Hence up ∈ F .
b) It is a direct consequence of normality and absolute monotonicity.
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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2.
Generalizations
Our main results are the following theorems which state the extensions of Hölder
and Minkowski inequalities. Taking F = `1 and k·kF = k·k1 in both Theorem
2.1 and Theorem 2.2, we get classical versions of these inequalities. Moreover, if we change the choices of F and k·kF then we can obtain many different
inequalities corresponding to these generalizations. Therefore, the following
results are quite productive.
Theorem 2.1. Let F be a sequence algebra and k·kF be an absolutely monotone
seminorm on F . Suppose u = (un ) , v = (vn ) ∈ F . Then
1/p
1/p
kuvkF ≤ kup kF kv q kF ,
where p > 1 and
1
p
+
1
q
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
= 1.
Proof. Assume that xn = |un |p and yn = |vn |q . It is immediate from Lemma
1.2(a) that x = (xn ) and y = (yn ) are members of F . Let M = kxkF and
N = kykF . Then it follows from inequality (1.1) that for each n,
x θ y 1−θ
xn
yn
n
n
≤θ
+ (1 − θ)
M
N
M
N
as 0 ≤ θ ≤ 1. Because k·kF is an absolutely monotone seminorm we write
x
xn θ yn 1−θ yn n
≤
θ
+
(1
−
θ)
.
M
N
M
N F
F
Hence
A Generalization of Hölder and
Minkowski Inequalities
θ 1−θ 1
xn yn ≤ 1,
F
M θ N 1−θ
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so that
θ 1−θ xn yn ≤ k(xn )kθ k(yn )k1−θ .
F
F
F
Setting θ = 1/p, we get
1/p 1/q xn yn ≤ k(xn )k1/p k(yn )k1/q ,
F
F
F
and putting xn = |un |p and yn = |vn |q , we obtain
1/p
1/q
k(|un vn |)kF ≤ k(|un |p )kF k(|vn |q )kF .
So, it follows from Lemma 1.2(b) that
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
kv q k1/q
.
kuvkF ≤ kup k1/p
F
F
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Theorem 2.2. Let F be a normal sequence algebra and k·kF be an absolutely
monotone seminorm on F . Then for every u = (un ) , v = (vn ) ∈ F and p ≥ 1,
1/p
1/p
1/p
k(u + v)p kF ≤ kup kF + kv p kF ,
p
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p
where (u + v) = ((un + vn ) ).
Proof. For p = 1, it is obvious.
Let p > 1. Proceeding with the manner of the proof in the classical version,
we write
(u + v)p = u (u + v)p−1 + v (u + v)p−1 .
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It follows from Theorem 2.1 that
1/q
1/q
1/p 1/p k(u + v)p kF ≤ kup kF (u + v)(p−1)q + kv p kF (u + v)(p−1)q F
F
1/q
1/p
1/p = kup kF + kv p kF (u + v)(p−1)q ,
F
where
k(u +
1 1
+ = 1. Hence, dividing the first and last terms by (u
p q
1/q
v)p kF , we obtain the inequality.
1/q
(p−1)q + v)
=
F
Example 2.1. Taking F = `∞ and k·kF = k·k∞ in both Theorem 2.1 and
Theorem 2.2, we obtain the inequalities
sup|un vn | ≤
n
p1 1q
q
sup|un |
· sup|vn |
p
n
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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n
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and
p
p1
sup|un + vn |
p1 p1
p
p
≤ sup|un |
+ sup|vn |
,
n
n
n
where u, v ∈ `∞ and p1 + 1q = 1 as p > 1. Hence, in fact, these elementary
inequalities are extended Hölder and Minkowski inequalities respectively.
Example 2.2. Now put F = `M and k·kF = k·kM in Theorem 2.1 and Theorem
2.2, where M satisfies the ∆´-condition. In this case, we write the inequalities
inf ρ > 0 :
X
M
|xk yk |
ρ
≤1
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p1
X |xk |p ≤ inf ρ > 0 :
M
≤1
ρ
1q
X |yk |q · inf ρ > 0 :
≤1
M
ρ
and
p1
X |xk + yk |p inf ρ > 0 :
M
≤1
ρ
p1
X |xk |p ≤ inf ρ > 0 :
M
≤1
ρ
p1
X |yk |p + inf ρ > 0 :
≤1
M
ρ
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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as Hölder and Minkowski inequalities respectively.
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3.
An Application
Now let us introduce the class F (X, λ, p) of vector-valued sequence spaces
which includes the space c0 (X, λ, p) investigated in [6] with some linear topological properties. Theorem 2.2 makes it possible to improve some topological
properties of the space F (X, λ, p).
Let F be an AK-BK normal sequence algebra such that the norm k·kF of
F is absolutely monotone and X be a seminormed space. Also suppose that
λ = (λk ) is a non-zero complex sequence and p = (pk ) is a sequence of strictly
positive real numbers. Define the vector-valued sequence class
F (X, λ, p) = {x ∈ s(X) : ([q (λk xk )]pk ) ∈ F } ,
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
where q is the seminorm of X and s(X) is the most general X-termed sequence
space. F (X, λ, p) becomes a linear space under natural co-ordinatewise vector
operations if and only if p ∈ `∞ (see Lascarides [1]). Taking F = c0 and X as
a Banach space we get the space c0 (X, λ, p) in [6].
Lemma 3.1. Let 0 < tk ≤ 1. If ak and bk are complex numbers then we have
|ak + bk |tk ≤ |ak |tk + |bk |tk
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[5, p.5].
Lemma 3.2. Let (X, q) be a seminormed space, and F a normal AK-BK space
with an absolutely monotone norm k·kF . Suppose p = (pk ) is a bounded sequence of positive real numbers. Then the map
n
X
x
en : [0, ∞) → [0, ∞) ; x
en (u) = [uq (λk xk )]pk ek k=1
F
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defined by means of x = (xk ) ∈ F (X, λ, p) and a positive integer n, is continuous, where (ek ) is a unit vector basis of F .
Proof. Since the norm function is continuous it is sufficient to show that the
mappings defined by
gk : [0, ∞) → F, gk (u) = [uqk (λk xk )]pk ek
are continuous. Let ui → 0 (i → ∞), then
gk (ui ) → (0, 0, . . .)
(i → ∞)
for each k. Hence, each gk is sequential continuous (it is equivalent to continuity
here).
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
Theorem 3.3. Define the function g : F (X, λ, p) −→ R by
1/M
g (x) = k([q (λk xk )]pk )kF
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,
where M = max (1, sup pn ). Then g is a paranorm on F (X, λ, p).
Proof. It is obvious that g(θ) = 0 and g(−x) = g(x). From the absolute monotonicity of k·kF , Lemma 3.1 and Theorem 2.2, we get
M 1/M
pk /M
g(x + y) = [q (λk xk + λk yk )]
F
M 1/M
pk /M
pk /M
≤ [q (λk xk )]
+ [q (λk yk )]
F
pk
≤ k([q (λk xk )]
= g(x) + g(y)
1/M
)kF
pk
+ k([q (λk yk )]
1/M
)kF
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for x, y ∈ F (X, λ, p).
To show the continuity of scalar multiplication assume that (µn ) is a sequence of scalars such that |µn − µ| → 0 (n → ∞) and g (xn − x) → 0 (n → ∞)
for an arbitrary sequence (xn ) ⊂ F (X, λ, p). We shall show that
g (µn xn − µx) → 0 (n → ∞) .
Say τn = |µn − µ| and we get
1/M
g (µn xn − µx) = k([q (λk (µn xnk − µxk ))]pk )kF
A Generalization of Hölder and
Minkowski Inequalities
1/M
= k([q (λk (µn xnk − µn xk + µn xk − µxk ))]pk )kF
n
1/M
(λk xk )] )kF
oM 1/M
pk /M
(λk (xnk
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
pk
− xk ))) + τn q
≤ k([|µ | q
n
pk /M
≤
+ [B (k, n)]
[A (k, n)]
,
F
(λk (xnk
where A (k, n) = Rq
− xk )), B (k, n) = τn q (λk xk ) and R =
n
max {1, sup |µ |}. Again by Theorem 2.2 we can write
1/M
1/M
g (µn xn − µx) ≤ k(A (k, n))kF + k(B (k, n))kF
pk 1/M
A
1/M
+ k(B (k, n))kF
≤ R
R
F
1/M
= Rg(xn − x) + k(B (k, n))kF
n
.
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1/M
(k, n))kF
Since g (x − x) → 0 (n → ∞) we must show that k(B
→ 0
(n → ∞). We can find a positive integer n0 such that 0 ≤ τn ≤ 1 for n ≥ n0 .
J. Ineq. Pure and Appl. Math. 7(5) Art. 193, 2006
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Say tk = [q (λk xk )]pk . Since t = (tk ) ∈ F and F is an AK-space, we get
m
∞
X
X
tk ek = [q (λk xk )]pk ek → 0 (m → ∞) ,
t −
k=1
F
k=m+1
F
where (ek ) is a unit vector basis of F . Therefore, for every ε > 0 there exists a
positive integer m0 such that
1
∞
X
M
ε
[q (λk xk )]pk ek < .
2
k=m +1
0
A Generalization of Hölder and
Minkowski Inequalities
F
For n ≥ n0 write [(τn q (λk xk ))]pk ≤ [f (q (λk xk ))]pk for each k. On the other
hand, we can write
1
1
∞
∞
X
M
X
M
ε
pk
pk
[τn q (λk xk )] ek ≤ [q (λk xk )] ek < .
2
k=m +1
k=m +1
0
0
F
F
Now, from Lemma 3.2, the function
m
0
X
x
em0 (u) = [(uq (λk xk ))]pk ek k=1
F
is continuous. Hence, there exists a δ (0 < δ < 1) such that
x
em0 (u) ≤
ε M
2
,
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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for 0 < u < δ. Also we can find a number ∆ such that τn < δ for n > ∆. So
for n > ∆ we have
m
0
X
ε
(e
xm0 (τn ))1/M = [τn q (λk xk )]pk ek < ,
2
k=1
F
and eventually we get
1/M
k([τn q (λk xk )]pk )kF
1
∞
X
M
pk
[τn q (λk xk )] ek =
k=1
F
1
m
∞
0
M
X
X
pk
pk
=
[τn q (λk xk )] ek +
[τn q (λk xk )] ek k=m0 +1
k=1
F
m
1
1
∞
0
X
M X
M
≤
[τn q (λk xk )]pk ek + [τn q (λk xk )]pk ek k=1
F
k=m0 +1
ε ε
< + = ε.
2 2
F
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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1/M
This shows that k(B (k, n))kF
Close
→ 0 (n → ∞).
Theorem 3.4. Let (X, q) be a complete seminormed space. Then F (X, λ, p) is
complete with the paranorm g. If X is a Banach space then F (X, λ, p) is an
FK-space, in particular, an AK-space.
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Proof. Let (xn ) be a Cauchy sequence in F (X, λ, p). Therefore
1/M
pk
g(xn − xm ) = k([q (λk (xnk − xm
k ))] )kF
→0
(m, n → ∞) ,
also, since F is an FK-space, for each k
pk
[q (λk (xnk − xm
→0
k ))]
(m, n → ∞)
and so |λk | q (xnk − xm
k ) → 0 (m, n → ∞). Because of the completeness of X,
there exists an xk ∈ X such that q (xnk − xk ) → 0 (n → ∞) for each k. Define
the sequence x = (xk ) with these points. Now we can determine a sequence
ηk ∈ c0 (0 < ηkn ≤ 1) such that
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
[|λk | q (xnk − xk )]pk ≤ ηkn [q (λk xnk )]pk
(3.1)
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since q
(xnk
− xk ) → 0. On the other hand,
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[q (λk xk )]pk ≤ D {[q (λk (xnk − xk ))]pk + [q (λk xnk )]pk } ,
where D = max 1, 2H−1 ; H = sup pk . From (3.1) we have
[q (λk xk )]pk ≤ D (1 + ηkn ) [q (λk xnk )]pk
≤ 2D [q (λk xnk )]pk .
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So we get x ∈ F (X, λ, p). Now, for each ε > 0 there exist n0 (ε) such that
[g(xn − xm )]M < εM
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for n, m > n0 .
J. Ineq. Pure and Appl. Math. 7(5) Art. 193, 2006
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Also, we may write from the AK-property of F that
m
∞
0
X
X
pk
m
m pk
n
n
[q (λk (xk − xk ))] ek ≤ [q (xk − xk )] ek k=1
k=1
F
F
n
= g(x − xm )M .
Letting m → ∞ we have
m
m
0
0
X
X
p
p
m
n
n
k
k
[q (λk (xk − xk ))] ek → [q (λk (xk − xk ))] ek k=1
F
<ε
k=1
M
F
for n > n0 .
Since (ek ) is a Schauder basis for F ,
m
0
X
[q (λk (xnk − xk ))]pk ek → k([q (λk (xnk − xk ))]pk )kF
k=1
F
< εM
as m0 → ∞.
Then we get g(xn − x) < ε for n > n0 so g(xn − x) → 0 (n → ∞).
For the rest of the theorem; we can say immediately that F (X, λ, p) is a
Frechet space, because X is a Banach space. Also, the projections
P̂k : F (X, λ, p) −→ X;
P̂k (x) = xk
are continuous since Pk = |λk | q ◦ P̂k for each k. Where Pk ’s are coordinate
mappings on F and they are continuous since F is an FK-space.
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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Let x[n] be the nth section of an element x of F (X, λ, p). We must prove
that x[n] → x in F (X, λ, p) for each x ∈ F (X, λ, p). Indeed,
g x − x[n] = g (0, 0, . . . , 0, xn+1 , xn+2 , . . .)
∞
X
pk
=
[q (λk xk )] ek → 0
k=n+1
F
since F is an AK-space. Hence F (X, λ, p) is an AK-space.
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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References
[1] C.G. LASCARIDES, A study of certain sequence spaces of Maddox and a
generalization of a theorem of Iyer, Pacific J. Math., 38 (1971), 487–500.
[2] J. LINDENSTRAUSS AND L. TZAFRIRI On Orlicz sequence space, Israel
J. Math., 10 (1971), 379–390.
[3] P.K. KAMTHAN AND M. GUPTA, Sequence Spaces and Series, Marcel
Dekker Inc., New York and Basel, 1981.
[4] E. KREYSZIG, Introductory Functional Analysis with Applications, John
Wiley & Sons, New York, 1978.
[5] S. NANDA AND B. CHOUDHARY, Functional Analysis with Applications,
John Wiley & Sons, New York, 1989.
[6] J. K. SRIVASTAVA AND B. K. SRIVASTAVA, Generalized Sequence
Space c0 (X, λ, p), Indian J. of Pure Appl. Math., 27(1) (1996), 73–84.
[7] A. WILANSKY, Modern Methods in Topological Vector Spaces, Mac-Graw
Hill, New York, 1978.
[8] A. WILANSKY, Summability Through Functional Analysis, Mathematics
Studies 85, North-Holland, Amsterdam, 1984.
A Generalization of Hölder and
Minkowski Inequalities
Yılmaz Yilmaz, M. Kemal Özdemir
and İhsan Solak
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J. Ineq. Pure and Appl. Math. 7(5) Art. 193, 2006
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