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 Poumai 2 1 Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110 007, India 2 Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi 110 021, India Correspondence should be addressed to S. K. Kaushik; shikk2003@yahoo.co.in 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], -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,  will denote a Banach space,  will denote a Hilbert space, let  ∗ the dual space of , [  ] be the closed linear span of {  } in the norm topology of , and let  be the canonical mapping of  into  ∗∗ . A series ∑ ∞    in a conjugate Banach space  ∗ is called weak-convergent to  if it converges in  ∗ , ∗∗ )-topology. In this case, we write   ∑ ∞    . A series ∑ ∞    in a conjugate Banach space  ∗ is called weak ∗ -convergent to  if it converges in  ∗ , )- topology. In this case, we write   ∗ ∑ ∞    .