Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 729235, 7 pages
doi:10.1155/2012/729235
Research Article
On Generalized Approximative
Properties of Systems
Turab Mursal Ahmedov and Zahira Vahid Mamedova
Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 9 B. Vaxabzade,
Baku AZ1141, Azerbaijan
Correspondence should be addressed to Zahira Vahid Mamedova, zahira eng13@hotmail.com
Received 5 January 2012; Accepted 15 February 2012
Academic Editor: Siegfried Gottwald
Copyright q 2012 T. Mursal Ahmedov and Z. Vahid Mamedova. This is an open access article
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Generalized concepts of b-completeness, b-independence, b-minimality, and b-basicity are
introduced. Corresponding concept of a space of coefficients is defined, and some of its properties
are stated.
1. Introduction
Theory of classical basis including Schauder basis is sufficiently well developed, and quite
a good number of monographs such as Day 1, Singer 2, 3, Young 4, Bilalov, Veliyev
5, Ch. Heil 6, and so forth have been dedicated to it so far. There are different versions
and generalizations of the concept of classical basis for more details see 2, 3. One of such
generalizations is proposed in 5, 7. In these works, the concept of bY -invariance is first
introduced and then used to obtain main results. It should be noted that the condition of
bY -invariance implies the fact that the corresponding mapping is tensorial.
In our work, we neglect the bY -invariance condition. We give more detailed consideration to the space of coefficients. We also state a criterion of basicity.
2. Needful Notations and Concepts
Let X, Y , Z be some Banach spaces, and let · X , · Y , · Z be the corresponding
norms. Assume that we are given some bounded bilinear mapping b : X × Y → Z, that
is, bx; yZ ≤ c xX yY , for all x ∈ X, for all y ∈ Y , where c is an absolute constant. For
simplicity, we denote xy ≡ bx; y. Let M ⊂ Y be some set. By Lb M we denote the b-span
of M. So, by definition, Lb M ≡ {z ∈ Z : ∃{xk }n1 ⊂ X, ∃{yk }n1 ⊂ M, z nk1 xk yk }. By · we
denote the closure in the corresponding space.
2
International Journal of Mathematics and Mathematical Sciences
System {yn }n∈N ⊂ Y is called b-linearly independent if ∞
n1 xn yn 0 in Z implies that
xn 0, for all n ∈ N.
In the context of the above mentioned, the concept of usual completeness is stated as
follows.
Definition 2.1. System {yn }n∈N ⊂ Y is called b-complete in Z if Lb {yn }n∈N ≡ Z. We will also
need the concepts of b-biorthogonal system and b-basis.
Definition 2.2. System {yn∗ }n∈N ⊂ LZ; X is called b-biorthogonal to the system {yn }n∈N ⊂ Y if
yn∗ xyk δnk x, for all n, k ∈ N, for all x ∈ X, where δnk is the Kronecker symbol.
Definition 2.3. System {yn }n∈N ⊂ Y is called b-basis in Z if for for all z ∈ Z, ∃!{xn }n∈N ⊂ X :
z ∞
n1 xn yn .
To obtain our main results we will use the following concept.
Definition 2.4. System {yn }n∈N ⊂ Y is called nondegenerate if ∃cn > 0 : xX ≤ cn xyn Z , for
all x ∈ X, for all n ∈ N.
3. Main Results
3.1. Space of Coefficients
Let {yn }n∈N ⊂ Y be some system. Assume that
Ky ≡
{xn }n∈N
∞
⊂ X : the series
xn yn converges in Z .
3.1
n1
With regard to the ordinary operations of addition and multiplication by a complex number,
Ky is a linear space. We introduce a norm · Ky in Ky as follows:
xKy
m
sup xn yn ,
m n1
3.2
Z
where x ≡ {xn }n∈N ∈ Ky . In fact, it is clear that
λ xKy |λ| xKy ,
x1 x2 Ky ≤ x1 Ky x2 Ky ,
∀λ ∈ C;
∀xk ∈ Ky , k 1, 2.
3.3
Assume that xKy 0 for some x ≡ {xn }n∈N ∈ Ky . Let n0 inf {n : xk 0, for all k ≤
n−1}. In what follows we will suppose that the system {yn }n∈N is nondegenerate. Let n0 < ∞.
We have
n0
m
1
sup xn yn ≥ xn yn xn0 yn0 Z ≥ xn0 X > 0.
cn
m
n1
n1
Z
Z
3.4
International Journal of Mathematics and Mathematical Sciences
3
But this is contrary to xKy 0. Consequently, n0 ∞, that is, x 0. Thus, Ky ; · Ky is a normed space. Let us show that it is complete. Let {xn }n∈N ⊂ Ky be some fundamental
n
sequence with xn ≡ {xk }k∈N ⊂ X. For arbitrary fixed number k ∈ N. We have
np np
n
n
yk xk − xk ≤ ck xk − xk
X
Z
k k−1 np
np
n
n
x − xi
yi −
xi − xi
yi ck i1 i
i1
Z
m np
n
xi − xi
yi ≤ 2ck sup
m i1
2ck xn − xnp Ky −→ 0,
3.5
as n, p −→ ∞.
n
n
As a result, for all fixed k ∈ N the sequence {xk }n∈N is fundamental in X. Let xk →
xk , n → ∞. Let us take arbitrary positive ε. Then ∃n0 : for all n ≥ n0 , for all p ∈ N, we
have xn − xnp Ky < ε. Thus
m np
n
x − xk
yk < ε,
k1 k
∀n ≥ n0 , ∀p, m ∈ N.
3.6
∀n ≥ n0 , ∀m ∈ N.
3.7
∀n ≥ n0 , ∀m, p ∈ N.
3.8
Z
Passing to the limit as p → ∞ yields
m n
x − xk yk ≤ ε,
k1 k
Z
It is easy to see that
mp
n
x − xk yk ≤ 2ε,
km k
Z
Since the series
we have
∞
k1
n
n
n
xk yk converges in Z, it is clear that ∃m0 : for all m ≥ m0 , for all p ∈ N,
mp
n
xk yk < ε.
km
3.9
Z
Then it follows from the previous inequality that
mp
mp
xk yk ≤ xn − xk yk k
km
Z
km
Z
n
∀m ≥ m0 ,
mp
n
xk yk ≤ 3ε,
km
∀p ∈ N.
Z
3.10
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International Journal of Mathematics and Mathematical Sciences
Consequently, the series ∞
k1 xk yk converges in Z, and therefore x ≡ {xk }k∈N ∈ Ky . From
3.7 it follows directly that xn − xKy → 0, n → ∞. Thus, Ky is a Banach space.
Let us consider the operator K : Ky → Z, defined by the expression
Kx ∞
n1
xn yn , x ≡ {xn }n∈N .
3.11
It is obvious that K is a linear operator. Let z Kx. We have
∞
m
K xZ zZ xn yn ≤ sup xn yn xKy .
n1
m n1
Z
3.12
Z
It follows that K ∈ LKy ; Z and K ≤ 1. Let x0 ≡ {x; 0; · · · }, x ∈ X. It is clear that
Kx0 Z x0 Ky . Consequently, K 1. It is absolutely obvious that if the system {yn }n∈N ⊂
Y is b-linearly independent, then KerK {0}. In this case, ∃K −1 : Z → Ky . If Im K is closed,
then, by the Banach’s theorem on the inverse operator, we obtain that K −1 ∈ LIm K; Ky .
The same considerations are valid in the case when the system {yn }n∈N has a b-biorthogonal
system. We will call operator K a coefficient operator. Thus, we have proved the following.
Theorem 3.1. Every nondegenerate system Sy ≡ {yn }n∈N ⊂ Y is corresponded by a Banach space
of coefficients Ky and coefficient operator K ∈ LKy ; Z, K 1. If the system Sy is b-linearly
independent or has a b-biorthogonal system, then ∃K −1 . Moreover, if Im K is closed, then K −1 ∈
LIm K; Ky .
In what follows, we will need the concept of b-basis in the space of coefficients Ky .
Definition 3.2. System {Tn }n∈N ⊂ LX; Ky is called b-basis in Ky if for for all x ∈
∞
Ky , ∃!{xn }n∈N ⊂ X : x n1 Tn xn convergence in Ky .
Consider the operators En : X −→ Ky : En x {δnk x}k∈N , n ∈ N. We have
En xKy xyn Z ≤ yn Y xX ,
∀x ∈ X.
3.13
Thus, En ∈ LX; Ky , for all n ∈ N. Take x ≡ {xn }n∈N ∈ Ky . Then
m
x − En xn n1
Ky
⎫
⎧
⎨
⎬
0; · · · ; 0; xm1 ; xm2 ; · · · ⎩
⎭
m
Ky
mp
sup xn yn −→ 0,
p nm1
3.14
z
as m → ∞, since the series ∞
n1 xn yn converges in Z. As a result, we obtain that x ∞
n1 En xn . Consider the operators Pn : Ky → X : Pn x xn , n ∈ N. We have
m
Pn xX xn X ≤ cn xn yn Z ≤ cn sup xk yk cn xKy .
m
k1
Z
3.15
International Journal of Mathematics and Mathematical Sciences
5
Consequently, Pn ∈ LKy ; X, n ∈ N. Let us show that the expansion x ∞
n1 En xn is unique.
∞
∞
∞
Let n1 En xn 0. We have 0 Pk n1 En xn n1 Pk En xn xk , for all k ∈ N. As a
result, we obtain that the system {En }n∈N forms a b-basis for Ky . We will call this system a
canonical system. So we have proved the following.
Theorem 3.3. Let Ky be a space of coefficients of nondegenerate system {yn }n∈N ⊂ Y . Then the
canonical system {En }n∈N forms a b-basis for Ky .
Let the nondegenerate system {yn }n∈N ⊂ Y form a b-basis for Z. Consider the coefficient
operator K : Ky → Z. By definition of b-basis, the equation K x z is solvable with regard
to x ∈ Ky for for all z ∈ Z. It is absolutely clear that Ker K {0}. Then it follows from
Theorem 3.1 and Banach theorem that K −1 ∈ LZ; Ky . Consequently, operator K performs
isomorphism between Ky and Z.
Vice versa, let {yn }n∈N ⊂ Y be a nondegenerate system and let Ky be a corresponding
space of coefficients. Assume that a coefficient operator K ∈ LKy ; Z is an isomorphism.
Take for all z ∈ Z. It is clear that ∃x ≡ {xn }n∈N ∈ Ky : Kx z, that is z ∞
n1 xn yn in Z.
Consequently, z can be expanded in a series with respect to the system {yn }n∈N . Let us show
0
0
0
that such an expansion is unique. Let ∞
n1 xn yn 0 for some x ≡ {xn }n∈N ∈ Ky . This means
0
0
0
that Kx 0 ⇒ x 0 ⇒ xn 0, for for all n ∈ N. Thus, the system {yn }n∈N forms a b-basis
for Z. So the following theorem is proved.
Theorem 3.4. Let Ky be a space of coefficients of nondegenerate system {yn }n∈N and let K be a
corresponding coefficient operator. Then this system forms a b-basis for Z if and only if K is an
isomorphism in LKy ; Z.
3.2. Criterion of Basicity
Let the systems {yn }n∈N ⊂ Y and {yn∗ }n∈N ⊂ LZ; X be b-biorthogonal. Take for all z ∈ Z and
consider the partial sums
Sn z n
k1
yk∗ zyk ,
n ∈ N.
3.16
We have
n
n
m
∗
∗
∗
Sn Sm z yk Sm zyk yk
yi zyi yk
k1
m
n k1 i1
k1
δki yi∗ zyk i1
min{n;m}
k1
yk∗ zyk Smin{n;m} z,
3.17
∀n, m ∈ N.
Hence, S2n Sn , for for all n ∈ N, that is, Sn is a projector in Z. It follows directly from the
estimate
∗
y zyk ≤ cy∗ z yk ≤ cy∗ yk z ,
Z
k
k
k
Y
Y
Z
X
∀z ∈ Z,
3.18
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International Journal of Mathematics and Mathematical Sciences
that Sn is a continuous projector. Suppose that the nondegenerate system {yn }n∈N ⊂ Y forms
a b-basis for Z. Then for for all z ∈ Z has a unique expansion z ∞
n1 xn yn in Z. We denote
the correspondence z → xn by yn∗ : yn∗ z xn , for all n ∈ N. It is obvious that yn∗ : Z → X is
a linear operator. Let Ky be a space of coefficients of basis {yn }n∈N , and let K : Ky → Z be the
corresponding coefficient operator. By Theorem 3.4, K is an isomorphism. We have
m
∗ yn z xn ≤ cn xn yn ≤ cn sup
x
y
cn xKy
k
k
X
X
m
k1
cn K −1 z
Ky
≤ cn K −1 zZ ,
Z
3.19
where x ≡ {xn }n∈N . Consequently, {yn∗ }n∈N ⊂ LZ; X. It follows directly from the uniqueness
of the expansion that yn∗ xyk δnk x; for all n, k ∈ N, for for all x ∈ X. As a result, we obtain
that the system {yn∗ }n∈N is b-biorthogonal to {yn }n∈N . Let us consider the projectors Sm ∈ LZ
for for all z ∈ Z:
Sm z m
yn∗ zyn ,
m ∈ N.
3.20
n1
As the series 3.20 converges for for all z ∈ Z, it follows from Banach-Steinhaus theorem that
M supSm < ∞.
3.21
m
It is absolutely obvious that the system {yn }n∈N is b-complete in Z. Thus, if the system {yn }n∈N
forms a b-basis for Z, then 1 it is b-complete in Z; 2 it has a b-biorthogonal system; 3 the
corresponding family of projectors is uniformly bounded.
Vice versa, let the system {yn }n∈N be b-complete in Z and have a b-biorthogonal system
{yn∗ }n∈N . Assume that the corresponding family of projectors {Sm }m∈N is uniformly bounded,
that is, relation 3.21 holds. Let z ∈ Z be an arbitrary element. Let us take arbitrary positive
m 0
m0
0
ε. It is clear that ∃{xn }m
n1 ⊂ X : z −
n1 xn yn Z < ε. Let z0 n1 xn yn . We have
yn∗ z0 m0
k1
yn∗ xk yk xn ,
n 1, m0 ; yn∗ z0 0, ∀n > m0 .
3.22
As a result, we obtain for m ≥ m0 that
m
z − Sm zZ ≤ z − z0 Z z0 − Sm zZ ≤ ε yn∗ z − z0 yn ≤ M 1 ε.
n1
3.23
Z
From the arbitrariness of ε we get limm → ∞ Sm z z. Consequently, for for all z ∈ Z can
be expanded in a series with respect to the system {yn }n∈N . The existence of b-biorthogonal
system implies the uniqueness of the expansion. Thus, the following theorem is valid.
International Journal of Mathematics and Mathematical Sciences
7
Theorem 3.5. Nondegenerate system {yn }n∈N ⊂ Y forms a b-basis for Z if and only if the following
conditions are satisfied:
1 the system is b-complete in Z;
2 it has a b-biorthogonal system {yn∗ }n∈N ⊂ LZ; X;
3 the family of projectors 3.21 is uniformly bounded.
References
1
2
3
4
5
6
7
M. M. Day, Normed Linear Spaces, Izdat. Inostr. Lit., Moscow, Russia, 1961.
I. Singer, Bases in Banach Spaces, vol. 1, Springer, New York, NY, USA, 1970.
I. Singer, Bases in Banach Spaces, vol. 2, Springer, New York, NY, USA, 1981.
R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, NY, USA, 1980.
B. T. Bilalov and S. G. Veliyev, Some problems of Basis, Elm, Baku, Azerbaijan, 2010.
Ch. Heil, A Basis Theory Primer, Springer, New York, NY, USA, 2011.
B. T. Bilalov and A. I. Ismailov, “Bases and tensor product,” Transactions of National Academy of Sciences
of Azerbaijan, vol. 25, no. 4, pp. 15–20, 2005.
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