Table Of Contentp-HAHN SEQUENCE SPACE
MURATKIRIŞCI
Abstract. Themainpurposeofthepresentpaperistointroducethespacehpandstudyofsomepropertiesof
newsequencespace. Alsowecomputetheirdualspacesandcharacterizations ofsomematrixtransformations.
4
1
0
2 1. Introduction
n By ω = CN, we denote the space of all real- or complex-valued sequences, where C denotes the complex
a field and N = {0,1,2,...}. Each linear subspace of ω is called a sequence space. For x = (x ) ∈ ω, we shall
J k
0 employ the sequence spaces ℓ∞ = {x : supk|xk| < ∞}, c = {x : limkxk exists }, c0 = {x : limkxk = 0},
bs = {x : sup | n x | < ∞}, cs = {x : ( n x ) ∈ c} and ℓ = {x : |x |p < ∞, 1 ≤ p < ∞}
1 n k=1 k k=1 k p k k
which are BanachPspace with the following norPms; kxkℓ∞ = supk|xk|, kxkbs =Pkxkcs = supn| nk=1xk| and
] kxk =( |x |p)1/p as usual, respectively. And also
A ℓp k k P
F P ∞
h. bvp = x=(xk)∈ω : |xk−xk−1|p <∞ ,
( )
t Xk=1
a
m λ = {x=(x )∈ω :(kx )∈λ}.
k k
[ Z
A sequence, whose k − th term is x , is denoted by x or (x ). A coordinate space (or K−space) is a
k k
1 vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a
v sequence space λ with a linear topology is called a K-space provided each of the maps p : λ → C defined by
5 i
p (x)=x iscontinuousforalli∈N. ABK−spaceisaK−space,whichisalsoaBanachspacewithcontinuous
7 i i
4 coordinate functionals fk(x)=xk, (k =1,2,...).A K−space λ is called an FK−space provided λ is a complete
2 linear metric space. An FK−space whose topology is normable is called a BK− space.If a normed sequence
1. spaceλcontainsa sequence(bn)with the propertythat foreveryx∈λ thereis unique sequenceofscalars(αn)
0 such that
4
1 lim kx−(α0b0+α1b1+...+αnbn)k=0
n→∞
:
v
then (b ) is calledSchauder basis (or briefly basis)for λ. The series α b which hasthe sum x is then called
n k k
i
X the expansion of x with respect to (b ), and written as x = α b . An FK−space λ is said to have AK
n k k
P
r property,if φ⊂λ and{ek} is a basis for λ, where ek is a sequence whose only non-zeroterm is a 1 in kth place
a for each k ∈N and φ=span{ek}, the set of all finitely non-zerPo sequences. If φ is dense in λ, then λ is called
an AD-space, thus AK implies AD.
Let λ and µ be two sequence spaces, and A = (a ) be an infinite matrix of complex numbers a , where
nk nk
k,n∈N. Then, we say that A defines a matrix mapping from λ into µ, and we denote it by writing A:λ→µ
if for every sequence x=(x )∈λ. The sequence Ax={(Ax) }, the A-transform of x, is in µ; where
k n
(1.1) (Ax) = a x for each n∈N.
n nk k
k
X
Forsimplicityinnotation,hereandinwhatfollows,thesummationwithoutlimitsrunsfrom0to∞. By(λ:µ),
we denote the class of all matrices A such that A : λ → µ. Thus, A ∈ (λ : µ) if and only if the series on the
right side of (1.1) convergesfor each n∈N and each x∈λ and we have Ax={(Ax)n}n∈N ∈µ for all x∈λ. A
sequence x is said to be A-summable to l if Ax converges to l which is called the A-limit of x.
The matrix domain λ of an infinite matrix A in a sequence space λ is defined by
A
(1.2) λ ={x=(x )∈ω :Ax∈λ}
A k
2010 Mathematics Subject Classification. Primary46A45;Secondary 46A45,46A35.
Key words and phrases. Matrix transformations, Hahn sequence space, BK-space, dual spaces, Schauder basis, AK-property,
AD-property.
Thisworkwassupported byScientificProjects Coordination UnitofIstanbul University. Projectnumber35565.
1
which is a sequence space(for several examples of matrix domains, see [2] p. 49-176). In [5], Başar and Altay
have defined the sequence space bv which consists of all sequences such that ∆-transforms of them are in ℓ
p p
where ∆ denotes the matrix ∆=(δ )
nk
(−1)n−k , (n−1≤k ≤n)
∆=δ =
nk 0 , (0≤k <n−1 or k >n)
(cid:26)
for all k,n∈N. The space [ℓ(p)]Au =bv(u,p) has been studied by Başar et al. [3] where
(−1)n−ku , (n−1≤k ≤n)
Au =au = k
nk 0 , (0≤k <n−1 or k >n)
(cid:26)
for all k,n∈N.
Inthepresentpaper,weintroducep−Hahnsequencespace. Weinvestigateitssomepropertiesandcompute
duals of this space and characterizedsome matrix transformations.
We assume throughout that p−1+q−1 = 1 for p,q ≥ 1. We denote the collection of all finite subsets of N
be F.
2. New Hahn Sequence Space
Hahn [7] introduced the BK−space h of all sequences x=(x ) such that
k
∞
h= x: k|∆x |<∞ and lim x =0 ,
k k
( k→∞ )
k=1
X
where ∆x =x −x , for all k ∈N. The following norm
k k k+1
kxk = k|∆x |+sup|x |
h k k
k
k
X
was defined on the space h by Hahn [7] (and also [6]). Rao ([11], Proposition 2.1) defined a new norm on h as
kxk= k|∆x |. Goes and Goes [6] proved that the space h is a BK−space.
k k
Hahn proved following properties of the space h:
P
Lemma 2.1. (i) h is a Banach space.
(ii) h⊂ℓ ∩ c .
1 0
(iii) hβ =σ∞.
R
In[6], GoesandGoesstudiedfunctionalanalyticpropertiesofthe BK−spacebv ∩dℓ . Additionally,Goes
0 1
and Goes consideredthe arithmetic means of sequences in bv and bv ∩dℓ , and used an important fact which
0 0 1
the sequence of arithmetic means (n−1 n x ) of an x ∈ bv is a quasiconvex null sequence. And also Goes
k=1 k 0
and Goes proved that h=ℓ ∩ bv =ℓ ∩ bv .
1 1 0
P
R R
Rao [11] studied some geometric properties of Hahn sequence space and gave the characterizationsof some
classes of matrix transformations.
Balasubramanian and Pandiarani[1] defined the new sequence space h(F) called the Hahn sequence space
of fuzzy numbers and proved that β− and γ−duals of h(F) is the Cesàro space of the set of all fuzzy bounded
sequences.
Kirişci [8] compiled to studies on Hahn sequence space and defined a new Hahn sequence space by Cesàro
mean in [9].
Now, we introduce the sequence space h by
p
∞
h = x: (k|∆x |)p <∞ and lim x =0 (1<p<∞)
p k k
( k→∞ )
k=1
X
where ∆x =(x −x ), (k =1,2,...). If we take p=1, h =h which called Hahn sequence spaces.
k k k+1 p
Define the sequence y = (y ), which will be frequently used, by the M-transform of a sequence x = (x ),
k k
i.e.,
(2.1) y =(Mx) =k(x −x ).
k k k k+1
2
where M =(m ) with
nk
n , (n=k)
(2.2) m = −n , (n+1=k)
nk
0 , other
for all k,n∈N.
Theorem 2.2. h =ℓ ∩ bvp =ℓ ∩ bvp
p p p 0
Proof. We consider R R
k∆x ≤x +∆(kx ).
k k k
Then, for x∈ℓ ∩ bvp
p
R n n n
k|∆x |≤ |x |+ |∆(kx )|
k k k
k=1 k=1 k=1
X X X
and from |a+b|p ≤2p(|a|p+|b|p),(1≤p<∞), we obtain
n n n
kp|∆x |p ≤2p |x |p+ |∆(kx )|p .
k k k
" #
k=1 k=1 k=1
X X X
For each positive integer r, we get
r r r
kp|∆x |p ≤2p |x |p+ |∆(kx )|p .
k k k
" #
k=1 k=1 k=1
X X X
and as r →∞
∞ ∞ ∞
kp|∆x |p ≤2p |x |p+ |∆(kx )|p .
k k k
" #
k=1 k=1 k=1
X X X
and limk→∞xk =0. Then x∈hp and
(2.3) ℓ ∩ bvp ⊂h .
p p
Z
Let x∈h and we consider
p
∞ ∞ ∞
|x |p− |∆(kx )|p ≤ kp|∆x |p.
k+1 k k
k=1 k=1 k=1
X X X
The the series ∞ |x |p is convergent from the definition of ℓ . Also ∞ |∆(kx )|p < ∞ and therefore
k=1 k+1 p k=1 k
x∈ℓ ∩ bvp. Then
p
P P
(2.4) R h ⊂ℓ ∩ bvp.
p p
Z
Form (2.3) and (2.4), we obtain h =ℓ ∩ bvp. (cid:3)
p p
Theorem 2.3. The sequence space h is a BKR-space with AK.
p
Proof. If x is any sequence, we write σ (x) =M x. Let ε> 0 and x ∈h be given. Then there exists N such
n n p
that
(2.5) |σ (x)|<ε/2
n
for all n≥N. Now let m≥N be given. Then we have for all n≥m+1 by (2.5)
∞ p 1/p
σ x−x[m] ≤ k(∆x ) ≤|σ (x)|+|σ (x)|<ε/2+ε/2=ε
n k n m
(cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12) "k=Xm+1(cid:12)(cid:12) (cid:12)(cid:12) #
whence kx−x(cid:12)[m]khp ≤ε for(cid:12)all m≥N. (cid:12)(cid:12)This sho(cid:12)(cid:12)ws x=limm→∞x[m]. (cid:3)
Since h is an AK-space and every AK-space is AD, we can give the following corollary:
p
Corollary 2.4. The sequence space h has AD.
p
3
Theorem 2.5. Define a sequence b(k) = b(k) of elements of the space h for every fixed k ∈N by
n n∈N p
(cid:8) (cid:9) 1 , (n≤k)
b(k) = k
n 0 , (n>k)
(cid:26)
Then the sequence b(k) is a basis for the space h , and any x∈h has a unique representation of the form
n n∈N p p
(2.6) (cid:8) (cid:9) x= λ b(k)
k
k
X
where λ =(Mx) for all k ∈N and 1≤p<∞.
k k
Proof. It is clear that {b(k)}⊂h , since
p
(2.7) Mb(k) =ek ∈ℓ , (k =0,1,2,...).
1
1≤p<∞. Let x∈h be given. For every non-negative integer m, we put
p
m
(2.8) x[m] = λ b(k).
k
k=0
X
Then, we obtain by applying M to (2.8) with (2.7) that
m m
Mx[m] = λ Mb(k) = (Mx) ek
k k
k=0 k=0
X X
and
0 , (0≤i≤m)
M(x−x[m]) = ; (i,m∈N).
(Mx) , (i>m)
i i
n o (cid:26)
Given ε>0, then there is an integer m such that
0
∞ 1/p
ε
|i.(∆x) |p <
i
2
" #
i=m
X
for all m≥m . Hence,
0
∞ 1/p ∞ 1/p
ε
kx−x[m]k = |i.(∆x) |p ≤ |i.(∆x) |p < <ε
hp i i 2
" # " #
iX=m iX=m0
for all m≥m which proves that x∈h is represented as in (2.6).
0 p
Toshow the uniqueness ofthis representation,we assumethatx= µ b(k). Now,we define the transfor-
k k
mation T with the notation of (2.1), from h to ℓ by x7→y =Tx. The linearity of T is clear. Since the linear
p p
P
transformation T is continuous we have at this stage that
(Mx) = µ {Mb(k)} = µ ek =µ ; (n∈N)
n k n k n n
k k
X X
which contradicts the fact that (Mx) = λ for all n ∈ N. Hence, the representation (2.6) of x ∈ h is
n n p
unique. (cid:3)
Theorem 2.6. Except the case p=2, the space h is not an inner product space, therefore not a Hilbert space
p
for 1<p<∞.
Proof. For p = 2, we will show that the space h is a Hilbert space. Since the space h is a BK-space from
2 p
Theorem2.3, the space h is a BK-space,for p=2. Also its normcanbe obtained froman inner product, i.e.,
2
kxk =hk∆x,k∆xi1/2 holds. Then the space h is a Hilbert space.
h2 2
Now consider the sequences e = (1,0,0,0,···) and e = (0,1,0,0,···). Then we see that ke +e k2 +
1 2 1 2 hp
ke −e k2 6= 2. ke k2 +ke k2 , i.e., the norm of the space h does not satisfy the parallelogram equality,
1 2 hp 1 hp 2 hp p
whichmenasthatthenormcannotbeobtainedfrominnerproduct. Hence,thespaceh withp6=2isaBanach
(cid:0) (cid:1) p
space that is not a Hilbert space. (cid:3)
Now, we give some inclusion relations concerning with the space h .
p
Theorem 2.7. Neither of the spaces hp and ℓ∞ includes the other one, where1<p<∞.
4
Proof. Now we choose the sequences a = (a ) and b = (b ) such that a = (a ) = {(−1)k} and b = (b ) =
k k k k
ki=11/(i+1). The sequence a=(ak) is in ℓ∞\hp andthe sequenceb=(bk)is in hp\ℓ∞. So,the sequences hp
and ℓ∞ does not include each other. (cid:3)
P
Theorem 2.8. If 1≤p<r, then h ⊂h .
p r
Proof. This can be obtained by analogy with the proof of Theorem 2.6 in [5]. So, we omit the details. (cid:3)
3. Duals of New Hahn Sequence Space
In this section, we state and prove the theorems determining the α-, β- and γ-duals of the sequence space
h .
p
Let x and y be sequences, X and Y be subsets of ω and A = (a )∞ be an infinite matrix of complex
nk n,k=0
numbers. We write xy = (x y )∞ , x−1∗Y = {a ∈ ω : ax ∈ Y} and M(X,Y) = x−1 ∗Y = {a ∈ ω :
k k k=0 x∈X
ax ∈ Y for all x ∈ X} for the multiplier space of X and Y. In the special cases of Y = {ℓ ,cs,bs}, we write
1
xα = x−1 ∗ℓ , xβ = x−1 ∗cs, xγ = x−1 ∗bs and Xα = M(X,ℓ ), Xβ = M(X,cs)T, Xγ = M(X,bs) for the
1 1
α−dual, β−dual, γ−dualof X. By A =(a )∞ we denote the sequence in the n−th row of A, and we write
n nk k=0
A (x)= ∞ a x n=(0,1,...) and A(x)=(A (x))∞ , provided A ∈xβ for all n.
n k=0 nk k n n=0 n
GivenPan FK−space X containing φ, its conjugate is denoted by X′ and its f−dual or sequential dual is
denoted by Xf and is given by Xf ={ all sequences (f(ek)):f ∈X′}.
Letλbeasequencespace. Thenλiscalledperfect ifλ=λαα;normal ify ∈λwhenever|y |≤|x |, k ≥1
k k
for some x∈λ; monotone if λ contains the canonical preimages of all its stepspace.
Lemma 3.1. (i). A∈(h:ℓ ) if and only if
1
∞
(3.1) |a | converges, (k =1,2,...)
nk
n=1
X
∞ k
1
(3.2) sup a <∞.
nυ
k
k n=1(cid:12)υ=1 (cid:12)
X(cid:12)X (cid:12)
(ii). A∈(ℓ :ℓ ) if and only if (cid:12) (cid:12)
p 1 (cid:12) (cid:12)
q
sup a <∞
nk
K∈F k (cid:12)n∈K (cid:12)
X(cid:12) X (cid:12)
(cid:12) (cid:12)
Lemma 3.2. (i). A∈(h:c) if and only if (cid:12) (cid:12)
k
1
(3.3) sup a <∞
nυ
k
n,k (cid:12)υ=1 (cid:12)
(cid:12)X (cid:12)
(cid:12) (cid:12)
(3.4) lim a e(cid:12)xists, (k(cid:12) =1,2,...)
nk
n→∞
(ii). A∈(ℓ :c) if and only if (3.4) holds and
p
q
(3.5) sup a <∞, 1<p<∞
nk
n
k
X(cid:12) (cid:12)
Lemma 3.3. (i). A∈(h:ℓ∞) if and on(cid:12)ly if(cid:12)(3.3) holds.
(ii). A∈(ℓp :ℓ∞) if and only if (3.5) holds with 1<p≤∞.
Lemma 3.4. A∈(h:c ) if and only if (3.3) holds and
0
(3.6) lim a =0
nk
n→∞
Lemma 3.5. A∈(h:h) if and only if (3.6) holds and
∞
(3.7) n|a −a | converges, (k =1,2,...)
nk n+1,k
n=1
X
∞ k
1
(3.8) sup n (a −a ) <∞.
nv n+1,v
k
k n=1 (cid:12)v=1 (cid:12)
X (cid:12)X (cid:12)
(cid:12) 5 (cid:12)
(cid:12) (cid:12)
Theorem 3.6. We define the sets d and d as follows:
1 2
q
1
d = {a=(a )∈ω : sup a <∞} 1<p<∞
1 k n
K∈FXk (cid:12)(cid:12)nX∈K k (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12)
d = {a=(a )∈ω : sup (cid:12) a (cid:12)<∞}.
2 k n
K∈FXk (cid:12)(cid:12)nX∈K k (cid:12)(cid:12)
(cid:12) (cid:12)
Then [h ]α =d and [h]α =d . (cid:12) (cid:12)
p 1 2 (cid:12) (cid:12)
Proof. We give the proof only for the case [h ]α =d . Let us take any a=(a )∈ω and consider the equation
p 1 k
∞
a
(3.9) a x = ny =(Dy) (n∈N)
n n j n
j
j=n
X
where D =(d ) is defined by
nk
an , k ≥n
d = k
nk 0 , k <n
(cid:26)
for all k,n∈N. It follows from (3.9) with Lemma 3.1(ii) that ax=(a x )∈ℓ whenever x=(x )∈h if and
n n 1 k p
only if Dy ∈ℓ whenever y =(y )∈ℓ . This means that a=(a )∈[h ]α whenever x=(x )∈h if and only
1 k p n p n p
if D ∈(h :ℓ ). This gives the result that [h ]α =d . (cid:3)
p 1 p 1
Hahn[7] proved that [h]β = σ∞ where σ∞ = {a = (ak) ∈ ω : supn n1| nk=1ak| < ∞}. We can give β-dual
of h .
p
P
Theorem 3.7. Let 1<p<∞. Then, [h ]β =d where
p 3
q
n
d = a=(a )∈ω :sup(n−1)q a <∞
3 k n∈N Xk (cid:12)(cid:12)(cid:12)Xj=k j(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
Proof. Consider the equation (cid:12)(cid:12) (cid:12)(cid:12)
n n n n k
y a
(3.10) a x = a j = j y =(By) (n∈N);
k k k k n
j k
k=1 k=1 j=k k=1 j=1
X X X X X
where B =(b ) are defined by
nk
k aj , (n≤k)
b = j=1 k
nk
0 , (n>k)
(cid:26) P
forallk,n∈N. ThuswededucefromLemma3.2(ii)with(3.10)thatax=(a x )∈cswheneverx=(x )∈h
k k k p
if and only if By ∈ c whenever y = (y ) ∈ ℓ . Thus, (a ) ∈ cs and (a ) ∈ d by (3.4) and (3.5), respectively.
k p k k 3
Nevertheless, the inclusion d ⊂cs holds and, thus, we have (a )∈d whence [h ]β =d . (cid:3)
3 k 3 p 3
Lemma 3.8. ([12], Theorem 7.2.7) Let X be an FK−space with X ⊃φ. Then,
(i) Xβ ⊂Xγ ⊂Xf;
(ii) If X has AK, Xβ =Xf;
(iii) If X has AD, Xβ =Xγ.
From Theorem 2.3, Corollary 2.4 and Lemma 3.8, we can write the following corollary:
Corollary 3.9. (i) [h ]β =[h ]f
p p
(ii) [h ]β =[h ]γ.
p p
Lemma 3.10. Let λ be a sequence space. Then the following assertions are true:
(i) λ is perfect ⇒ λ is normal ⇒ λ is monotone;
(ii) λ is normal ⇒ λα =λγ;
(iii) λ is monotone ⇒ λα =λβ.
Combining Theorem 3.6, Theorem 3.7 and Lemma 3.10, we can give the following corollary:
Corollary 3.11. The space h is not monotone and so it is neither normal nor perfect.
p
6
4. Matrix Transformations
In this section, we characterize some matrix transformations on the space h .
p
Lemma 4.1. [5] Let λ,µ be any two sequence spaces, A be an infinite matrix and U a triangle matrix ma-
trix.Then, A∈(λ:µ ) if and only if UA∈(λ:µ).
U
If we define a = n(a −a ), then we can give following corollary from Lemma 4.1 with U = M
nk nk n+1,k
defined by (2.2):
Corollary 4.2. e (i) A∈(ℓ :h) if and only if
1
sup |a |<∞
nk
k n
X
(ii) A∈(c:h)=(c0 :h)=(ℓ∞ :h) if and only ief
sup a <∞
nk
K∈FXn (cid:12)(cid:12)kX∈K (cid:12)(cid:12)
(cid:12) (cid:12)
Theorem 4.3. Suppose that the entries of the in(cid:12)finiteema(cid:12)trices A = (a ) and E = (e ) are connected with
(cid:12) (cid:12) nk nk
the relation
(4.1) e =a
nk nk
for all k,n ∈ N, where a = ∞ anj and µ be any sequence space. Then A ∈ (h : µ) if and only if
nk j=k j p
{ank}k∈N ∈[hp]β for all n∈N anPd E ∈(h:µ).
Proof. Let µ be any given sequence spaces. Suppose that (4.1) holds between A = (a ) and E = (e ), and
nk nk
take into account that the spaces h and h are norm isomorphic.
p
Let A ∈ (hp : µ) and take any y = yk ∈ h. Then EM exists and {ank}k∈N ∈ [hp]β which yields that
{enk}k∈N ∈ℓ1 for each n∈N. Hence, Ey exists and thus
e y = a x
nk k nk k
k k
X X
for all n∈N. We have that Ey =Ax which leads us to the consequence E ∈(h:µ).
Conversely, let {ank}k∈N ∈ d1 for all n ∈ N and E ∈ (h : µ) hold, and take any x = xk ∈ hp. Then, Ax
exists. Therefore, we obtain from the equality
∞
a
nj
a x = y
nk k k
j
k k j=k
X X X
for all n∈N. Thus Ax=Ey and this shows that A∈(h :µ). (cid:3)
p
If we use the Corollary 4.2 and change the roles of the spaces h with µ in Theorem ref4thm1, we can give
p
following theorem:
Theorem 4.4. Suppose that the entries of the infinite matrices A = (a ) and A = (a ) are connected with
nk nk
the relation a =n(a −a ) for all k,n∈N and µ be any sequence space. Then A∈(µ:h ) if and only
nk nk n+1,k p
if and A∈(µ:h). e e
e
Proof. Let z =(z )∈µ and consider the following equality
e k
m m
a z = n(a −a )z for all, m,n∈N
nk k nk n+1,k k
k=0 k=0
X X
which yields that as m→∞ theat (Az) ={M(Az)} for all n∈N. Therefore, one can observe from here that
n n
Az ∈h whenever z ∈µ if and only if Az ∈h whenever z ∈µ. (cid:3)
p
e
We can give following corollaries from Lemma 3.1-3.5, Corollary 4.2, Theorem 4.3 and Theorem 4.4:
e
Corollary 4.5. (i) A∈(hp :ℓ∞) if and only if {ank}k∈N ∈[hp]β for all n∈N and
q
k
1
(4.2) sup a <∞
nv
k k (cid:12) (cid:12)!
(cid:12)Xv=1 (cid:12)
(cid:12) (cid:12)
(cid:12) 7 (cid:12)
(cid:12) (cid:12)
(ii) A∈(hp :c) if and only if {ank}k∈N ∈[hp]β, (4.2) holds and
(4.3) lim a =α (k ∈N)
nk k
n→∞
(iii) A∈(hp :c0) if and only if {ank}k∈N ∈[hp]β, (4.2) holds and (4.3) holds with αk =0.
(iv) A∈(hp :ℓ1) if and only if {ank}k∈N ∈[hp]β and
∞
|a |q converges, (k =1,2,...)
nk
n=1
X
∞ k q
1
sup a <∞.
kq nυ
k n=1(cid:12)υ=1 (cid:12)
X(cid:12)X (cid:12)
Corollary 4.6. (i) A∈(ℓ:h ) if and only if (cid:12) (cid:12)
p (cid:12) (cid:12)
sup a <∞
nk
K∈FXk (cid:12)(cid:12)nX∈K (cid:12)(cid:12)
(cid:12) (cid:12)
(ii) A∈(c:hp)=(c0 :hp)=(ℓ∞ :hp) if and(cid:12)only eif (cid:12)
(cid:12) (cid:12)
sup a <∞
nk
K∈FXn (cid:12)(cid:12)kX∈K (cid:12)(cid:12)
(cid:12) (cid:12)
5. (cid:12)Concelu(cid:12)sion
(cid:12) (cid:12)
Hahn [7] defined the space h and gave some properties. Goes and Goes [6] studied its different properties.
Rao [11] introduced the Hahn sequence space and investigated some properties in Banach space theory. Kir-
işci[8]compiledtostudiesofHahnsequencespaceanddefinedanewHahnsequencespacebyCesàromeanin[9].
In this paper, we defined the space p−Hahn sequence spaces and gave some properties. In section 3, we
computethedualsofthespaceh andcharacterizesomematrixtransformationsrelatedtothisspace,insection
p
4.
Finally, we should note that, as a naturalcontinuationof the presentpaper,one canstudy the paranormed
Hahn sequence space. Also it can be obtained the new Hahn sequence space by using Euler mean, Riesz mean,
generalized weighted mean etc.
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Department of Mathematical Education, Hasan Ali Yücel Education Faculty, Istanbul University, Vefa,
34470, Fatih,Istanbul, Turkey
E-mail address: [email protected], [email protected]
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