Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 318659, 4 pages http://dx.doi.org/10.1155/2013/318659
Research Article On Schauder Frames in Conjugate Banach Spaces S. K. Kaushik,1 S. K. Sharma,1 and Khole Timothy Poumai2 1 2
Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110 007, India Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi 110 021, India
Correspondence should be addressed to S. K. Kaushik;
[email protected] Received 31 August 2012; Accepted 17 November 2012 Academic Editor: Ding-Xuan Zhou Copyright © 2013 S. K. Kaushik et al. is 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. Weak∗ -Schauder frame in conjugate Banach spaces has been introduced and studied. A sufficient condition for the existence of weak∗ -Schauder frame in the conjugate space of a separable Banach space has been given. It has been shown that ℓ∞ has weak∗ Schauder frame. Finally, a sufficient condition for the existence of a Schauder frame sequence has been given.
1. Introduction Frames for Hilbert spaces were formally introduced by Duffin and Schaeffer [1]. Later, Daubechies et al. [2] found new applications to wavelets in which frames played an important role. Frames are main tools for use in signal and image processing, compression, sampling theory, optics, �lter banks, signal detection, and so forth. In order to have more applications of frames, several notions generalizing the concept of frames have been introduced and studied, namely, pseudoframes [3], oblique frames [4], frames of subspaces (fusion frames) [5], 𝐺𝐺-frames [6], orthogonal frames [7, 8], and so forth. Feichtinger and Gröchenig [9] extended the notion of atomic decomposition to Banach spaces. Gröchenig [10] introduced a more general concept for Banach spaces called Banach frame. Banach frames and atomic decompositions were further studied in [11–14]. Han and Larson [15] de�ned Schauder frame for a Banach space. In [16], Casazza, et al. gave various de�nitions of frames for Banach spaces including that of Schauder frame. In 2008, Casazza et al. [17] studied the coefficient quantization of Schauder frames in Banach spaces. Liu [18] gave the concepts of minimal and maximal associated bases with respect to Schauder frames and closely connected them with the duality theory. In [19], Liu and Zheng gave a characterization of Schauder frames which are near-Schauder bases. In fact, they generalized some results due to Holub [20]. Beanland et al. [21] proved that the upper and lower estimates theorems for �nite dimensional
decompositions of Banach spaces can be extended and modi�ed to Schauder frames and gave a complete characterization of duality for Schauder frames. Φ-Schauder frames were introduced and studied by Vashisht [22]. Recently, Liu [23] associated an operator with a Schauder frame and called it Hilbert-Schauder frame operator. In the present paper, we introduce the concept of weak∗ Schauder frame and weak-Schauder frame in conjugate Banach spaces. A sufficient condition for the existence of weak∗ -Schauder frame in a conjugate Banach space of a separable Banach space has been given. Also, an example of a conjugate space of a nonseparable Banach space which has no weak∗ -Schauder frame is given. Further, it has been shown that ℓ∞ has weak∗ -Schauder frame. Finally, a sufficient condition for the existence of a Schauder frame sequence has been given.
2. Preliminaries roughout this paper, 𝐸𝐸 will denote a Banach space, 𝐻𝐻 will denote a Hilbert space, let 𝐸𝐸∗ the dual space of 𝐸𝐸, [𝑥𝑥𝑛𝑛 ] be the closed linear span of {𝑥𝑥𝑛𝑛 } in the norm topology of 𝐸𝐸, and let 𝜋𝜋 be the canonical mapping of 𝐸𝐸 into 𝐸𝐸∗∗ . A series ∑∞ 𝑛𝑛𝑛𝑛 𝑓𝑓𝑖𝑖 in a conjugate Banach space 𝐸𝐸∗ is called weak-convergent to 𝑓𝑓 if it converges in 𝜎𝜎𝜎𝜎𝜎∗ , 𝐸𝐸∗∗ )-topology. In this case, we write 𝑤𝑤 ∗ ∞ 𝑓𝑓 = ∑∞ 𝑛𝑛𝑛𝑛 𝑓𝑓𝑛𝑛 . A series ∑𝑛𝑛𝑛𝑛 𝑓𝑓𝑖𝑖 in a conjugate Banach space 𝐸𝐸 ∗ ∗ is called weak -convergent to 𝑓𝑓 if it converges in 𝜎𝜎𝜎𝜎𝜎 , 𝐸𝐸𝐸𝑤𝑤∗
topology. In this case, we write 𝑓𝑓 = ∑∞ 𝑛𝑛𝑛𝑛 𝑓𝑓𝑛𝑛 .
2
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�e�nition 1. A sequence {𝑥𝑥𝑛𝑛 }𝑛𝑛𝑛𝑛 ⊂ 𝐻𝐻 is called a frame for 𝐻𝐻 if there exist 𝐴𝐴, 𝐵𝐵 with 0 < 𝐴𝐴 𝐴 𝐴𝐴𝐴 𝐴 such that ∞
2
𝐴𝐴‖𝑥𝑥‖2 ≤ 𝑥𝑥𝑥 𝑥𝑥𝑛𝑛 ≤ 𝐵𝐵‖𝑥𝑥‖2 , 𝑛𝑛𝑛𝑛
𝑥𝑥 𝑥𝑥𝑥𝑥
(1)
e positive constants 𝐴𝐴 and 𝐵𝐵, respectively, are called lower and upper frame bounds of the frame {𝑥𝑥𝑛𝑛 }𝑛𝑛𝑛𝑛 . e inequality (1) is called the frame inequality.
�e�nition �. Let 𝐸𝐸 be a Banach space. A sequence {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 ({𝑥𝑥𝑛𝑛 } ⊂ 𝐸𝐸𝐸𝐸𝐸𝐸𝑛𝑛 } ⊂ 𝐸𝐸∗ ) is called a Schauder frame for 𝐸𝐸 if ∞
𝑥𝑥 𝑥 𝑓𝑓𝑛𝑛 (𝑥𝑥) 𝑥𝑥𝑛𝑛 , 𝑛𝑛𝑛𝑛
𝑥𝑥 𝑥𝑥𝑥𝑥
(2)
In view of eorem 6, one may observe that if ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇 is an exact retro Banach frame for 𝐸𝐸∗ , then there exists a sequence {𝑔𝑔𝑛𝑛 } in 𝐸𝐸∗ , called an admissible sequence to the retro Banach frame ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇, such that 𝑔𝑔𝑖𝑖 (𝑥𝑥𝑗𝑗 ) = 𝛿𝛿𝑖𝑖𝑖𝑖 , for all 𝑖𝑖, 𝑗𝑗𝑗𝑗.
3. Main Results
�e begin with the following de�nitions of weak∗ -Schauder frame and weak-Schauder frame in 𝐸𝐸∗ .
�e�nition �. Let 𝐸𝐸 be a Banach space, let {𝑓𝑓𝑛𝑛 } be a sequence in 𝐸𝐸∗ and let {Φ𝑛𝑛 } be a sequence in 𝐸𝐸∗∗ , and let 𝑀𝑀 be a subset of 𝐸𝐸∗ . en, {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is said to be (a) weak-Schauder frame for 𝐸𝐸∗ with respect to 𝑀𝑀 if 𝑤𝑤
∞
𝑓𝑓 = Φ𝑛𝑛 𝑓𝑓 𝑓𝑓𝑛𝑛 ,
𝑓𝑓 𝑓𝑓𝑓𝑓
(5)
𝑓𝑓 𝑓𝑓𝑓𝑓
(6)
�e�nition �. Let 𝐸𝐸 be a Banach space. A sequence {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 ({𝑥𝑥𝑛𝑛 } ⊂ 𝐸𝐸𝐸𝐸𝐸𝐸𝑛𝑛 } ⊂ 𝐸𝐸∗ ) is called a Schauder frame sequence for 𝐸𝐸 if {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a Schauder frame for [𝑥𝑥𝑛𝑛 ].
(b) weak∗ -Schauder frame for 𝐸𝐸∗ with respect to 𝑀𝑀 if
�e�nition �. Let 𝐸𝐸 be a Banach space and let 𝐸𝐸∗ be its conjugate space. Let (𝐸𝐸∗ )𝑑𝑑 be a Banach space of scalar-valued sequences associated with 𝐸𝐸∗ , indexed by ℕ. Let {𝑥𝑥𝑛𝑛 } ⊂ 𝐸𝐸 and 𝑇𝑇 𝑇 𝑇𝑇𝑇∗ )𝑑𝑑 → 𝐸𝐸∗ be given. e pair ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇 is called a retro Banach frame for 𝐸𝐸∗ with respect to (𝐸𝐸∗ )𝑑𝑑 if
In particular, if 𝑀𝑀 𝑀𝑀𝑀∗ , then {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is simply called a weak-Schauder frame and 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤∗ -Schauder frame for 𝐸𝐸∗ , respectively. Further, one may observe that if {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a Schauder frame for 𝐸𝐸∗ , then it is also a weak∗ -Schauder frame (weak-Schauder frame) for 𝐸𝐸∗ .
Next, we give the de�nition of a retro Banach frame introduced in [24].
(i) {𝑓𝑓𝑓𝑓𝑓𝑛𝑛 )} ∈ (𝐸𝐸∗ )𝑑𝑑 , 𝑓𝑓 𝑓𝑓𝑓∗ ,
(ii) there exist positive constants 𝐴𝐴 and 𝐵𝐵 with 0 < 𝐴𝐴 𝐴 𝐵𝐵 𝐵𝐵 such that 𝐴𝐴𝑓𝑓𝐸𝐸∗ ≤ 𝑓𝑓 𝑥𝑥𝑛𝑛 (𝐸𝐸∗ )𝑑𝑑 ≤ 𝐵𝐵𝑓𝑓𝐸𝐸∗ ,
𝑓𝑓 𝑓𝑓𝑓∗ ,
(iii) 𝑇𝑇 is a bounded linear operator such that 𝑇𝑇 𝑓𝑓 𝑥𝑥𝑛𝑛 = 𝑓𝑓𝑓
𝑓𝑓 𝑓𝑓𝑓∗ .
(3)
(4)
e positive constants 𝐴𝐴 and 𝐵𝐵, respectively, are called lower and upper frame bounds of the retro Banach frame ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇. e operator 𝑇𝑇 𝑇 𝑇𝑇𝑇∗ )𝑑𝑑 → 𝐸𝐸∗ is called the reconstruction operator (or the preframe operator). e inequality (3) is called the retro frame inequality. A retro Banach frame ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑛𝑛 } ⊂ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸∗ )𝑑𝑑 → 𝐸𝐸∗ ) for 𝐸𝐸 with respect to (𝐸𝐸∗ )𝑑𝑑 with bounds 𝐴𝐴, 𝐵𝐵 is said to be tight, if it is possible to choose 𝐴𝐴 𝐴𝐴𝐴, normalized tight, if 𝐴𝐴 𝐴𝐴𝐴𝐴𝐴, and exact, if there exists no reconstruction operator 𝑇𝑇0 such that ({𝑥𝑥𝑛𝑛 }𝑛𝑛 𝑛 𝑛𝑛 , 𝑇𝑇0 ) (𝑗𝑗𝑗𝑗𝑗 is a retro Banach frame 𝐸𝐸∗ . Finally, we give the following results which will be used in the subsequent results. ∗
eorem 5 (see [25]). If {𝑔𝑔𝑛𝑛 } ⊂ (ℓ∞ ) and lim𝑛𝑛 𝑛𝑛 𝑔𝑔𝑛𝑛 (𝑥𝑥𝑥𝑥 0, for all 𝑥𝑥 𝑥𝑥∞ , then lim𝑛𝑛 𝑛𝑛 Φ(𝑔𝑔𝑛𝑛 ) = 0 for all Φ ∈ 𝐸𝐸∗∗ .
eorem 6 (see [24]). Let ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑛𝑛 } ⊂ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸∗ )𝑑𝑑 → 𝐸𝐸∗ ) be a retro Banach frame for 𝐸𝐸∗ with respect to (𝐸𝐸∗ )𝑑𝑑 . en, ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇 is exact if and only if 𝑥𝑥𝑛𝑛 ∉ [𝑥𝑥𝑖𝑖 ]𝑖𝑖 𝑖𝑖𝑖 , for all 𝑛𝑛 𝑛 𝑛.
𝑤𝑤∗
𝑛𝑛𝑛𝑛 ∞
𝑓𝑓 = Φ𝑛𝑛 𝑓𝑓 𝑓𝑓𝑛𝑛 , 𝑛𝑛𝑛𝑛
Now, we give an example of a conjugate space of a nonseparable Banach space which has no weak∗ -Schauder frame.
Example 8. ℓ∗∞ has no weak∗ -Schauder frame. Assume on the contrary that ℓ∗∞ has a weak∗ -Schauder frame {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 ({𝑓𝑓𝑛𝑛 } ⊂ ℓ∗∞ , {Φ𝑛𝑛 } ⊂ ℓ∗∗ ∞ ). en, we have ∞
that is,
𝑓𝑓 (𝑥𝑥) = Φ𝑖𝑖 𝑓𝑓 𝑓𝑓𝑖𝑖 (𝑥𝑥) , 𝑖𝑖𝑖𝑖
𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥∗∞ ,
𝑛𝑛
𝑓𝑓 (𝑥𝑥) − Φ𝑖𝑖 𝑓𝑓 𝑓𝑓𝑖𝑖 (𝑥𝑥) ⟶ 0, 𝑖𝑖𝑖𝑖
So, by eorem 5, we have 𝑛𝑛
lim Φ 𝑓𝑓 𝑓 Φ𝑛𝑛 𝑓𝑓 𝑓𝑓𝑖𝑖 = 0, 𝑛𝑛 𝑛𝑛 𝑖𝑖𝑖𝑖
(7)
as 𝑛𝑛 𝑛𝑛𝑛
(8)
Φ ∈ ℓ∗∗ ∞.
(9)
is gives ℓ∗∞ is weak-separable. is is not possible. Hence, ℓ∗∞ has no weak∗ -Schauder frame. In view of Example 8 we have the following problem.
Problem 1. Does the conjugate space 𝐸𝐸∗ of a separable Banach space 𝐸𝐸 possess a weak∗ -Schauder frame?
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3
In this direction, we have the following result. Proposition 9. If a Banach space 𝐸𝐸 has a Schauder frame, then its conjugate Banach space has a weak∗ -Schauder frame. Proof. Let {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 be a Schauder frame for 𝐸𝐸; then ∞
𝑥𝑥 𝑥 𝑓𝑓𝑖𝑖 (𝑥𝑥) 𝑥𝑥𝑖𝑖 , 𝑖𝑖𝑖𝑖𝑖
erefore, for each 𝑓𝑓 𝑓𝑓𝑓∗ , we have
𝑥𝑥 𝑥𝑥𝑥𝑥
∞
𝑓𝑓 (𝑥𝑥) = 𝑓𝑓 𝑓𝑓𝑛𝑛 (𝑥𝑥) 𝑥𝑥𝑛𝑛 ∞
𝑖𝑖𝑖𝑖
(11)
= 𝜋𝜋 𝑥𝑥𝑛𝑛 𝑓𝑓 𝑓𝑓𝑛𝑛 (𝑥𝑥) , 𝑛𝑛𝑛𝑛
(10)
∗
𝑥𝑥 𝑥𝑥𝑥𝑥
Proof . It follows from Proposition 9.
Note. ℓ∞ does not have a Schauder frame.
Towards the converse of Proposition 9, we have the following result.
eorem 11. Let 𝐸𝐸 be a Banach space. Let {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 be a weak∗ -Schauder frame for 𝐸𝐸∗ such that each Φ𝑛𝑛 is weak∗ continuous. en, there exists a sequence {𝑥𝑥𝑛𝑛 } ⊂ 𝐸𝐸 such that {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a Schauder frame for 𝐸𝐸. ∞
𝑤𝑤∗
Φ𝑛𝑛 𝑓𝑓 𝑓𝑓𝑛𝑛 = 𝑓𝑓𝑓
𝑛𝑛𝑛𝑛
(12)
∗
Since each Φ𝑛𝑛 is weak -continuous, there exists a sequence {𝑥𝑥𝑛𝑛 } ⊂ 𝐸𝐸 such that Φ𝑛𝑛 = 𝜋𝜋𝜋𝜋𝜋𝑛𝑛 ), 𝑛𝑛 𝑛 𝑛. So, we have ∞
𝑓𝑓 (𝑥𝑥) = 𝜋𝜋 𝑥𝑥𝑛𝑛 𝑓𝑓 𝑓𝑓𝑛𝑛 (𝑥𝑥) 𝑛𝑛𝑛𝑛
is gives
∞
= 𝑓𝑓 𝑓𝑓𝑛𝑛 (𝑥𝑥) 𝑥𝑥𝑛𝑛 , 𝑛𝑛𝑛𝑛
∞
𝑥𝑥 𝑥 𝑓𝑓𝑛𝑛 (𝑥𝑥) 𝑥𝑥𝑛𝑛 , 𝑛𝑛𝑛𝑛
(13)
𝑓𝑓 𝑓𝑓𝑓∗ , 𝜋𝜋 (𝑥𝑥) ∈ 𝜋𝜋 (𝐸𝐸)
𝑛𝑛𝑛𝑛
(15)
if and only if {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak-Schauder frame for 𝐸𝐸∗ with respect to 𝜋𝜋𝜋𝜋𝜋𝜋.
Remark 13. Let 𝐸𝐸 be a re�exive Banach space. Let {𝑓𝑓𝑛𝑛 } be a sequence in 𝐸𝐸∗ and {Φ𝑛𝑛 } let be a sequence in 𝐸𝐸∗∗ . en, by eorem 12, {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak∗ -Schauder frame for 𝐸𝐸∗ if and only if {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak-Schauder frame for 𝐸𝐸∗ .
Finally, we give a sufficient condition for the existence of a Schauder frame sequence in a Banach space.
Proof. For each 𝑛𝑛 𝑛 𝑛, de�ne 𝑆𝑆𝑛𝑛 ∶ 𝐸𝐸 𝐸 𝐸𝐸 by 𝑛𝑛
𝑆𝑆𝑛𝑛 (𝑥𝑥) = 𝑓𝑓𝑖𝑖 (𝑥𝑥) 𝑥𝑥𝑖𝑖 ,
𝑥𝑥 𝑥𝑥𝑥𝑥
𝑖𝑖𝑖𝑖
Let 𝑆𝑆∗𝑛𝑛 be the adjoint operator to 𝑆𝑆𝑛𝑛 . en 𝑛𝑛
𝑛𝑛
𝑆𝑆∗𝑛𝑛 𝑔𝑔 (𝑥𝑥) = 𝑔𝑔 𝑓𝑓𝑖𝑖 (𝑥𝑥) 𝑥𝑥𝑖𝑖 = 𝑔𝑔 𝑥𝑥𝑖𝑖 𝑓𝑓𝑖𝑖 (𝑥𝑥) , 𝑖𝑖𝑖𝑖
is gives 𝑛𝑛
∗
𝑖𝑖𝑖𝑖
𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 ,
𝑆𝑆∗𝑛𝑛 𝑔𝑔 = 𝑔𝑔 𝑥𝑥𝑖𝑖 𝑓𝑓𝑖𝑖 , 𝑖𝑖𝑖𝑖
(16)
(17)
𝑛𝑛 𝑛𝑛𝑛 𝑛𝑛 𝑛 𝑛 𝑛
𝑔𝑔 𝑔𝑔𝑔∗ , 𝑛𝑛 𝑛𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛 𝑛
(18)
us, for every �nite linear combination 𝑔𝑔 𝑔 𝑔𝑚𝑚 𝑖𝑖𝑖𝑖 𝛽𝛽𝑖𝑖 𝑓𝑓𝑖𝑖 , we have 𝑛𝑛
𝑆𝑆∗𝑛𝑛 𝑔𝑔 = 𝑔𝑔 𝑥𝑥𝑖𝑖 𝑓𝑓𝑖𝑖 𝑖𝑖𝑖𝑖 𝑚𝑚
𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥∗ . 𝑥𝑥 𝑥𝑥𝑥𝑥
∞
(𝜋𝜋 (𝑥𝑥)) 𝑓𝑓 = Φ𝑛𝑛 𝑓𝑓 𝜋𝜋 (𝑥𝑥) 𝑓𝑓𝑛𝑛 ,
eorem 14. Let 𝐸𝐸 be a Banach space and let ({𝑥𝑥𝑛𝑛 }, 𝑇𝑇𝑇 be an exact retro Banach frame for 𝐸𝐸∗ with admissible sequence {𝑓𝑓𝑛𝑛 } ⊂ 𝐸𝐸∗ . en, {(𝑥𝑥𝑛𝑛 , 𝑓𝑓𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a Schauder-frame sequence in 𝐸𝐸.
Corollary 10. ℓ∞ has weak -Schauder frame.
Proof. For each 𝑓𝑓 𝑓𝑓𝑓∗ , we have
Proof. {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak∗ -Schauder frame for 𝐸𝐸∗ if and only if
= 𝛽𝛽𝑖𝑖 𝑓𝑓𝑖𝑖
(19)
𝑖𝑖𝑖𝑖
(14)
In the following result, we characterize weak∗ -Schauder frame in terms of weak-Schauder frame with respect to a subset. eorem 12. Let 𝐸𝐸 be a Banach space. Let {𝑓𝑓𝑛𝑛 } be a sequence in 𝐸𝐸∗ and let {Φ𝑛𝑛 } be a sequence in 𝐸𝐸∗∗ . en, {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak∗ -Schauder frame for 𝐸𝐸∗ if and only if {(𝑓𝑓𝑛𝑛 , Φ𝑛𝑛 )}𝑛𝑛𝑛𝑛 is a weak-Schauder frame for 𝐸𝐸∗ with respect to 𝜋𝜋𝜋𝜋𝜋𝜋.
= 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑔 𝑔𝑔𝑔𝑔
Let 𝑓𝑓 𝑓 𝑓𝑓𝑓𝑛𝑛 ] and 𝜀𝜀 𝜀 𝜀 be given. en, there exists a �nite 𝑚𝑚𝜀𝜀 linear combination 𝑔𝑔 𝑔 𝑔𝑗𝑗𝑗𝑗 𝛽𝛽(𝜀𝜀𝜀 𝑗𝑗 𝑓𝑓𝑗𝑗 such that ‖𝑓𝑓𝑓𝑓𝑓𝑓 𝑓 𝑓𝑓𝑓𝑓𝑓𝑓𝑓 1), where 𝜈𝜈𝜈 𝜈𝜈𝜈1≤𝑛𝑛𝑛𝑛 ‖𝑠𝑠𝑛𝑛 ‖ < ∞. Also, 𝑠𝑠∗𝑛𝑛 𝑓𝑓 − 𝑓𝑓 ≤ 𝑆𝑆∗𝑛𝑛 𝑓𝑓 − 𝑆𝑆∗𝑛𝑛 𝑔𝑔 + 𝑆𝑆∗𝑛𝑛 𝑔𝑔 − 𝑔𝑔 + 𝑔𝑔 𝑔𝑔𝑔
