# On Generalized Approximative Properties of Systems

Authors: Turab Mursal Ahmedov; Zahira Vahid Mamedova
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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.


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## Body

## 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 thebY-invariance condition. We give more detailed consideration to the space of coefficients. We also state a criterion of basicity.


## 2. Needful Notations and Concepts

LetX, 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, ∥b(x;y)∥Z≤c∥x∥X∥y∥Y,  for all x∈X, for all y∈Y, where c is an absolute constant. For simplicity, we denote xy≡b(x;y). Let M⊂Y be some set. By Lb[M] we denote the b-spanof M. So, by definition, Lb[M]≡{z∈Z:∃{xk}1n⊂X,∃{yk}1n⊂M,z=∑k=1nxkyk}. By (·¯) we denote the closure in the corresponding space.System{yn}n∈ℕ⊂Y is called b-linearlyindependent if ∑n=1∞xnyn=0inZ 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-biorthogonalsystem and b-basis.Definition 2.2.
System{yn*}n∈ℕ⊂L(Z;X) is called b-biorthogonalto the system {yn}n∈ℕ⊂Y if yn*(xyk)=δnkx,  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-basisin Z if for for all z∈Z,∃!{xn}n∈ℕ⊂X:z=∑n=1∞xnyn.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:∥x∥X≤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(3.1)Ky¯≡{{xn}n∈N⊂X:theseries∑n=1∞xnynconvergesinZ}.
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:(3.2)‖x¯‖Ky¯=sup⁡m‖∑n=1mxnyn‖Z,
where x¯≡{xn}n∈ℕ∈𝒦y¯. In fact, it is clear that(3.3)‖λx¯‖Ky¯=|λ|‖x¯‖Ky¯,∀λ∈C;‖x¯1+x¯2‖Ky¯≤‖x¯1‖Ky¯+‖x¯2‖Ky¯,∀x¯k∈Ky¯,k=1,2.
Assume that ∥x¯∥𝒦y¯=0 for some x¯≡{xn}n∈ℕ∈𝒦y¯.Letn0=inf{n:xk=0,forallk≤n-1}.In what follows we will suppose that the system {yn}n∈ℕ is nondegenerate.Let n0<+∞. We have
(3.4)sup⁡m‖∑n=1mxnyn‖Z≥‖∑n=1n0xnyn‖Z=‖xn0yn0‖Z≥1cn‖xn0‖X>0.
But this is contrary to ∥x¯∥𝒦y¯=0.Consequently,n0=+∞,that is,x¯=0.Thus,(𝒦y¯;∥·∥𝒦y¯) is a normed space. Let us show that it is complete. Let {x¯n}n∈ℕ⊂𝒦y¯ be some fundamental sequence with x¯n≡{xk(n)}k∈ℕ⊂X. For arbitrary fixed number k∈ℕ. We have (3.5)‖xk(n)-xk(n+p)‖X≤ck‖(xk(n)-xk(n+p))yk‖Z=ck‖∑i=1k(xi(n)-xi(n+p))yi-∑i=1k-1(xi(n)-xi(n+p))yi‖Z≤2cksup⁡m‖∑i=1m(xi(n)-xi(n+p))yi‖=2ck‖x¯n-x¯n+p‖Ky¯⟶0,asn,p⟶∞.
As a result, for all fixed k∈ℕ the sequence {xk(n)}n∈ℕ is fundamental in X. Let xk(n)→xk,n→∞. Let us take arbitrary positive ɛ. Then ∃n0: for all n≥n0, for all p∈ℕ, we have ∥x¯n-x¯n+p∥𝒦y¯<ɛ. Thus(3.6)‖∑k=1m(xk(n)-xk(n+p))yk‖Z<ɛ,∀n≥n0,∀p,m∈N.
Passing to the limit as p→∞ yields(3.7)‖∑k=1m(xk(n)-xk)yk‖Z≤ɛ,∀n≥n0,∀m∈N.
It is easy to see that(3.8)‖∑k=mm+p(xk(n)-xk)yk‖Z≤2ɛ,∀n≥n0,∀m,p∈N.
Since the series ∑k=1∞xk(n)yk converges in Z, it is clear that ∃m0(n): for all m≥m0(n), for all p∈ℕ, we have(3.9)‖∑k=mm+pxk(n)yk‖Z<ɛ.
Then it follows from the previous inequality that(3.10)‖∑k=mm+pxkyk‖Z≤‖∑k=mm+p(xk(n)-xk)yk‖Z+‖∑k=mm+pxk(n)yk‖Z≤3ɛ,∀m≥m0(n),∀p∈N.
Consequently, the series ∑k=1∞xkyk converges in Z, and therefore x¯≡{xk}k∈ℕ∈𝒦y¯. From (3.7) it follows directly that ∥x¯n-x¯∥𝒦y¯→0,n→∞. Thus,𝒦y¯ is a Banach space.Let us consider the operatorK:𝒦y¯→Z,defined by the expression
(3.11)Kx¯=∑n=1∞xnyn,x¯≡{xn}n∈N.
It is obvious that K is a linear operator.Let z=Kx¯. We have(3.12)‖Kx¯‖Z=‖z‖Z=‖∑n=1∞xnyn‖Z≤sup⁡m‖∑n=1mxnyn‖Z=‖x¯‖Ky¯.
It follows that K∈L(𝒦y¯;Z) and ∥K∥≤1. Let x¯0≡{x;0;⋯},x∈X. It is clear that ∥Kx¯0∥Z=∥x¯0∥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 ImK is closed, then, by the Banach’s theorem on the inverse operator, we obtain that K-1∈L(ImK;𝒦y¯). The same considerations are valid in the case when the system {yn}n∈ℕ has a b-biorthogonalsystem. We will call operator K a coefficient operator. Thus, we have proved the following.Theorem 3.1.
Every nondegenerate systemSy¯≡{yn}n∈ℕ⊂Y is corresponded by a Banach space of coefficients 𝒦y¯ and coefficient operator K∈L(𝒦y¯;Z),  ∥K∥=1.  If the system Sy¯ is  b-linearly  independent or has a  b-biorthogonal  system, then ∃K-1.  Moreover, if ImK is closed, then K-1∈L(ImK;𝒦y¯).In what follows, we will need the concept ofb-basisin the space of coefficients 𝒦y¯.Definition 3.2.
System{Tn}n∈ℕ⊂L(X;𝒦y¯) is called b-basis in 𝒦y¯ if for forallx¯∈𝒦y¯,∃!{xn}n∈ℕ⊂X:x¯=∑n=1∞Tnxn (convergence in 𝒦y¯).Consider the operatorsEn:X→𝒦y¯:Enx={δnkx}k∈ℕ,n∈ℕ. We have (3.13)‖Enx‖Ky¯=‖xyn‖Z≤‖yn‖Y‖x‖X,∀x∈X.
Thus, En∈L(X;𝒦y¯), for all n∈ℕ. Take x¯≡{xn}n∈ℕ∈𝒦y¯. Then (3.14)‖x¯-∑n=1mEnxn‖Ky¯=‖{0;⋯;0;︸mxm+1;xm+2;⋯}‖Ky¯=sup⁡p⁡‖∑n=m+1m+pxnyn‖z⟶0,
as m→∞, since the series ∑n=1∞xnyn converges in Z. As a result, we obtain that x¯=∑n=1∞Enxn. Consider the operators Pn:𝒦y¯→X:Pnx¯=xn,n∈ℕ. We have(3.15)‖Pnx¯‖X=‖xn‖X≤cn‖xnyn‖Z≤cnsup⁡m‖∑k=1mxkyk‖Z=cn‖x¯‖Ky¯.
Consequently, Pn∈L(𝒦y¯;X),n∈ℕ. Let us show that the expansion x¯=∑n=1∞Enxn is unique. Let ∑n=1∞Enxn=0. We have 0=Pk(∑n=1∞Enxn)=∑n=1∞Pk(Enxn)=xk,for all k∈ℕ. As a result, we obtain that the system {En}n∈ℕ forms a b-basisfor 𝒦y¯. We will call this system a canonical system. So we have proved the following.Theorem 3.3.
Let𝒦y¯ 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 forZ.Consider the coefficient operator K:𝒦y¯→Z. By definition of b-basis, the equation Kx¯=z is solvable with regard to x¯∈𝒦y¯ for for all z∈Z. It is absolutely clear that KerK={0}. Then it follows from Theorem 3.1 and Banach theorem that K-1∈L(Z;𝒦y¯). Consequently, operator Kperforms isomorphism between 𝒦y¯andZ.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∈L(𝒦y¯;Z) is an isomorphism. Take for all z∈Z. It is clear that ∃x¯≡{xn}n∈ℕ∈𝒦y¯:Kx¯=z,thatisz=∑n=1∞xnyninZ. Consequently, z can be expanded in a series with respect to the system {yn}n∈ℕ. Let us show that such an expansion is unique. Let ∑n=1∞xn0yn=0 for some x¯0≡{xn0}n∈ℕ∈𝒦y¯.  This means that Kx¯0=0⇒x¯0=0⇒x¯n0=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.
Let𝒦y¯ 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 L(𝒦y¯;Z).


### 3.2. Criterion of Basicity

Let the systems{yn}n∈ℕ⊂Y and {yn*}n∈ℕ⊂L(Z;X) be b-biorthogonal. Take for all  z∈Z and consider the partial sums
(3.16)Snz=∑k=1nyk*(z)yk,n∈N.
We have
(3.17)Sn(Smz)=∑k=1nyk*(Smz)yk=∑k=1nyk*[∑i=1myi*(z)yi]yk=∑k=1n∑i=1mδkiyi*(z)yk=∑k=1min⁡{n;m}yk*(z)yk=Smin⁡{n;m}z,∀n,m∈N.
Hence, Sn2=Sn, for for all  n∈ℕ, that is, Sn is a projector in Z.It follows directly from the estimate(3.18)‖yk*(z)yk‖Z≤c‖yk*(z)‖X‖yk‖Y≤c‖yk*‖‖yk‖Y‖z‖Z,∀z∈Z,
that Sn is a continuous projector. Suppose that the nondegenerate system {yn}n∈ℕ⊂Y forms a b-basisfor Z. Then for for all z∈Z has a unique expansion z=∑n=1∞xnyn 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¯ bea space of coefficientsof basis {yn}n∈ℕ, and let K:𝒦y¯→Z be the corresponding coefficient operator. By Theorem 3.4, K is an isomorphism. We have(3.19)‖yn*(z)‖X=‖xn‖X≤cn‖xnyn‖≤cnsup⁡m‖∑k=1mxkyk‖Z=cn‖x¯‖Ky¯=cn‖K-1z‖Ky¯≤cn‖K-1‖‖z‖Z,
where x¯≡{xn}n∈ℕ. Consequently, {yn*}n∈ℕ⊂L(Z;X). It follows directly from the uniqueness of the expansion that yn*(xyk)=δnkx; for all n,k∈ℕ, for for all x∈X. As a result, we obtain that the system {yn*}n∈ℕ is b-biorthogonalto {yn}n∈ℕ. Let us consider the projectors Sm∈L(Z) for for all z∈Z:(3.20)Smz=∑n=1myn*(z)yn,m∈N.
As the series (3.20) converges for for all  z∈Z, it follows from Banach-Steinhaus theorem that(3.21)M=sup⁡m‖Sm‖<+∞.
It is absolutely obvious that the system {yn}n∈ℕ isb-completein Z. Thus, if the system {yn}n∈ℕ forms a b-basisfor Z, then (1) it is b-completein Z; (2) it has a b-biorthogonalsystem; (3) the corresponding family of projectors is uniformly bounded.Vice versa, let the system{yn}n∈ℕ be b-complete inZ 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 ɛ. It is clear that ∃{xn}n=1m0⊂X:∥z-∑n=1m0xnyn∥Z<ɛ. Let z0=∑n=1m0xnyn. We have(3.22)yn*(z0)=∑k=1m0yn*(xkyk)=xn,n=1,m0¯;yn*(z0)=0,∀n>m0.
As a result, we obtain for m≥m0 that(3.23)‖z-Smz‖Z≤‖z-z0‖Z+‖z0-Smz‖Z≤ɛ+‖∑n=1myn*(z-z0)yn‖Z≤(M+1)ɛ.
From the arbitrariness of ɛ we get limm→∞Smz=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-biorthogonalsystem implies the uniqueness of the expansion. Thus, the following theorem is valid.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 isb-complete  in Z;(2)
it has ab-biorthogonal  system {yn*}n∈ℕ⊂L(Z;X);(3)
the family of projectors (3.21) is uniformly bounded.


## 3.1. Space of Coefficients

Let{yn}n∈ℕ⊂Y be some system. Assume that(3.1)Ky¯≡{{xn}n∈N⊂X:theseries∑n=1∞xnynconvergesinZ}.
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:(3.2)‖x¯‖Ky¯=sup⁡m‖∑n=1mxnyn‖Z,
where x¯≡{xn}n∈ℕ∈𝒦y¯. In fact, it is clear that(3.3)‖λx¯‖Ky¯=|λ|‖x¯‖Ky¯,∀λ∈C;‖x¯1+x¯2‖Ky¯≤‖x¯1‖Ky¯+‖x¯2‖Ky¯,∀x¯k∈Ky¯,k=1,2.
Assume that ∥x¯∥𝒦y¯=0 for some x¯≡{xn}n∈ℕ∈𝒦y¯.Letn0=inf{n:xk=0,forallk≤n-1}.In what follows we will suppose that the system {yn}n∈ℕ is nondegenerate.Let n0<+∞. We have
(3.4)sup⁡m‖∑n=1mxnyn‖Z≥‖∑n=1n0xnyn‖Z=‖xn0yn0‖Z≥1cn‖xn0‖X>0.
But this is contrary to ∥x¯∥𝒦y¯=0.Consequently,n0=+∞,that is,x¯=0.Thus,(𝒦y¯;∥·∥𝒦y¯) is a normed space. Let us show that it is complete. Let {x¯n}n∈ℕ⊂𝒦y¯ be some fundamental sequence with x¯n≡{xk(n)}k∈ℕ⊂X. For arbitrary fixed number k∈ℕ. We have (3.5)‖xk(n)-xk(n+p)‖X≤ck‖(xk(n)-xk(n+p))yk‖Z=ck‖∑i=1k(xi(n)-xi(n+p))yi-∑i=1k-1(xi(n)-xi(n+p))yi‖Z≤2cksup⁡m‖∑i=1m(xi(n)-xi(n+p))yi‖=2ck‖x¯n-x¯n+p‖Ky¯⟶0,asn,p⟶∞.
As a result, for all fixed k∈ℕ the sequence {xk(n)}n∈ℕ is fundamental in X. Let xk(n)→xk,n→∞. Let us take arbitrary positive ɛ. Then ∃n0: for all n≥n0, for all p∈ℕ, we have ∥x¯n-x¯n+p∥𝒦y¯<ɛ. Thus(3.6)‖∑k=1m(xk(n)-xk(n+p))yk‖Z<ɛ,∀n≥n0,∀p,m∈N.
Passing to the limit as p→∞ yields(3.7)‖∑k=1m(xk(n)-xk)yk‖Z≤ɛ,∀n≥n0,∀m∈N.
It is easy to see that(3.8)‖∑k=mm+p(xk(n)-xk)yk‖Z≤2ɛ,∀n≥n0,∀m,p∈N.
Since the series ∑k=1∞xk(n)yk converges in Z, it is clear that ∃m0(n): for all m≥m0(n), for all p∈ℕ, we have(3.9)‖∑k=mm+pxk(n)yk‖Z<ɛ.
Then it follows from the previous inequality that(3.10)‖∑k=mm+pxkyk‖Z≤‖∑k=mm+p(xk(n)-xk)yk‖Z+‖∑k=mm+pxk(n)yk‖Z≤3ɛ,∀m≥m0(n),∀p∈N.
Consequently, the series ∑k=1∞xkyk converges in Z, and therefore x¯≡{xk}k∈ℕ∈𝒦y¯. From (3.7) it follows directly that ∥x¯n-x¯∥𝒦y¯→0,n→∞. Thus,𝒦y¯ is a Banach space.Let us consider the operatorK:𝒦y¯→Z,defined by the expression
(3.11)Kx¯=∑n=1∞xnyn,x¯≡{xn}n∈N.
It is obvious that K is a linear operator.Let z=Kx¯. We have(3.12)‖Kx¯‖Z=‖z‖Z=‖∑n=1∞xnyn‖Z≤sup⁡m‖∑n=1mxnyn‖Z=‖x¯‖Ky¯.
It follows that K∈L(𝒦y¯;Z) and ∥K∥≤1. Let x¯0≡{x;0;⋯},x∈X. It is clear that ∥Kx¯0∥Z=∥x¯0∥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 ImK is closed, then, by the Banach’s theorem on the inverse operator, we obtain that K-1∈L(ImK;𝒦y¯). The same considerations are valid in the case when the system {yn}n∈ℕ has a b-biorthogonalsystem. We will call operator K a coefficient operator. Thus, we have proved the following.Theorem 3.1.
Every nondegenerate systemSy¯≡{yn}n∈ℕ⊂Y is corresponded by a Banach space of coefficients 𝒦y¯ and coefficient operator K∈L(𝒦y¯;Z),  ∥K∥=1.  If the system Sy¯ is  b-linearly  independent or has a  b-biorthogonal  system, then ∃K-1.  Moreover, if ImK is closed, then K-1∈L(ImK;𝒦y¯).In what follows, we will need the concept ofb-basisin the space of coefficients 𝒦y¯.Definition 3.2.
System{Tn}n∈ℕ⊂L(X;𝒦y¯) is called b-basis in 𝒦y¯ if for forallx¯∈𝒦y¯,∃!{xn}n∈ℕ⊂X:x¯=∑n=1∞Tnxn (convergence in 𝒦y¯).Consider the operatorsEn:X→𝒦y¯:Enx={δnkx}k∈ℕ,n∈ℕ. We have (3.13)‖Enx‖Ky¯=‖xyn‖Z≤‖yn‖Y‖x‖X,∀x∈X.
Thus, En∈L(X;𝒦y¯), for all n∈ℕ. Take x¯≡{xn}n∈ℕ∈𝒦y¯. Then (3.14)‖x¯-∑n=1mEnxn‖Ky¯=‖{0;⋯;0;︸mxm+1;xm+2;⋯}‖Ky¯=sup⁡p⁡‖∑n=m+1m+pxnyn‖z⟶0,
as m→∞, since the series ∑n=1∞xnyn converges in Z. As a result, we obtain that x¯=∑n=1∞Enxn. Consider the operators Pn:𝒦y¯→X:Pnx¯=xn,n∈ℕ. We have(3.15)‖Pnx¯‖X=‖xn‖X≤cn‖xnyn‖Z≤cnsup⁡m‖∑k=1mxkyk‖Z=cn‖x¯‖Ky¯.
Consequently, Pn∈L(𝒦y¯;X),n∈ℕ. Let us show that the expansion x¯=∑n=1∞Enxn is unique. Let ∑n=1∞Enxn=0. We have 0=Pk(∑n=1∞Enxn)=∑n=1∞Pk(Enxn)=xk,for all k∈ℕ. As a result, we obtain that the system {En}n∈ℕ forms a b-basisfor 𝒦y¯. We will call this system a canonical system. So we have proved the following.Theorem 3.3.
Let𝒦y¯ 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 forZ.Consider the coefficient operator K:𝒦y¯→Z. By definition of b-basis, the equation Kx¯=z is solvable with regard to x¯∈𝒦y¯ for for all z∈Z. It is absolutely clear that KerK={0}. Then it follows from Theorem 3.1 and Banach theorem that K-1∈L(Z;𝒦y¯). Consequently, operator Kperforms isomorphism between 𝒦y¯andZ.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∈L(𝒦y¯;Z) is an isomorphism. Take for all z∈Z. It is clear that ∃x¯≡{xn}n∈ℕ∈𝒦y¯:Kx¯=z,thatisz=∑n=1∞xnyninZ. Consequently, z can be expanded in a series with respect to the system {yn}n∈ℕ. Let us show that such an expansion is unique. Let ∑n=1∞xn0yn=0 for some x¯0≡{xn0}n∈ℕ∈𝒦y¯.  This means that Kx¯0=0⇒x¯0=0⇒x¯n0=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.
Let𝒦y¯ 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 L(𝒦y¯;Z).


## 3.2. Criterion of Basicity

Let the systems{yn}n∈ℕ⊂Y and {yn*}n∈ℕ⊂L(Z;X) be b-biorthogonal. Take for all  z∈Z and consider the partial sums
(3.16)Snz=∑k=1nyk*(z)yk,n∈N.
We have
(3.17)Sn(Smz)=∑k=1nyk*(Smz)yk=∑k=1nyk*[∑i=1myi*(z)yi]yk=∑k=1n∑i=1mδkiyi*(z)yk=∑k=1min⁡{n;m}yk*(z)yk=Smin⁡{n;m}z,∀n,m∈N.
Hence, Sn2=Sn, for for all  n∈ℕ, that is, Sn is a projector in Z.It follows directly from the estimate(3.18)‖yk*(z)yk‖Z≤c‖yk*(z)‖X‖yk‖Y≤c‖yk*‖‖yk‖Y‖z‖Z,∀z∈Z,
that Sn is a continuous projector. Suppose that the nondegenerate system {yn}n∈ℕ⊂Y forms a b-basisfor Z. Then for for all z∈Z has a unique expansion z=∑n=1∞xnyn 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¯ bea space of coefficientsof basis {yn}n∈ℕ, and let K:𝒦y¯→Z be the corresponding coefficient operator. By Theorem 3.4, K is an isomorphism. We have(3.19)‖yn*(z)‖X=‖xn‖X≤cn‖xnyn‖≤cnsup⁡m‖∑k=1mxkyk‖Z=cn‖x¯‖Ky¯=cn‖K-1z‖Ky¯≤cn‖K-1‖‖z‖Z,
where x¯≡{xn}n∈ℕ. Consequently, {yn*}n∈ℕ⊂L(Z;X). It follows directly from the uniqueness of the expansion that yn*(xyk)=δnkx; for all n,k∈ℕ, for for all x∈X. As a result, we obtain that the system {yn*}n∈ℕ is b-biorthogonalto {yn}n∈ℕ. Let us consider the projectors Sm∈L(Z) for for all z∈Z:(3.20)Smz=∑n=1myn*(z)yn,m∈N.
As the series (3.20) converges for for all  z∈Z, it follows from Banach-Steinhaus theorem that(3.21)M=sup⁡m‖Sm‖<+∞.
It is absolutely obvious that the system {yn}n∈ℕ isb-completein Z. Thus, if the system {yn}n∈ℕ forms a b-basisfor Z, then (1) it is b-completein Z; (2) it has a b-biorthogonalsystem; (3) the corresponding family of projectors is uniformly bounded.Vice versa, let the system{yn}n∈ℕ be b-complete inZ 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 ɛ. It is clear that ∃{xn}n=1m0⊂X:∥z-∑n=1m0xnyn∥Z<ɛ. Let z0=∑n=1m0xnyn. We have(3.22)yn*(z0)=∑k=1m0yn*(xkyk)=xn,n=1,m0¯;yn*(z0)=0,∀n>m0.
As a result, we obtain for m≥m0 that(3.23)‖z-Smz‖Z≤‖z-z0‖Z+‖z0-Smz‖Z≤ɛ+‖∑n=1myn*(z-z0)yn‖Z≤(M+1)ɛ.
From the arbitrariness of ɛ we get limm→∞Smz=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-biorthogonalsystem implies the uniqueness of the expansion. Thus, the following theorem is valid.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 isb-complete  in Z;(2)
it has ab-biorthogonal  system {yn*}n∈ℕ⊂L(Z;X);(3)
the family of projectors (3.21) is uniformly bounded.



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*Source: 729235-2012-05-27.xml*