The Kaczmarz algorithm is a versatile and computationally efficient method of reconstructing vectors in a Hilbert space using inner products against a sequence {e_n}. If the algorithm successfully reconstructs any vector in the space, we say that {en} is an effective sequence. Kwapien and Mycielski provide a twofold criterion for sequences to be effective. Expanding on these results, Haller and Szwarc present an extensive list of conditions equivalent to {e_n} being effective.
Within the context of a Hilbert space, we develop a dualized version of the Kaczmarz algorithm which is naturally suited for extension to a Banach space. We provide necessary and sufficient conditions for one part of the Kwapien-Mycielski criterion to be weakly satisfied, ensuring the dense weak convergence of the dualized algorithm in both Banach and Hilbert space. Furthermore, we show that the equivalences of Haller and Szwarc fail in the dualized context, instead separating into two sets of equivalent conditions. We conclude with a presentation of convergence conditions for periodic sequences in a finite-dimensional Banach space
