On Approximative Atomic Decompositions in Banach Spaces

Approximative atomic decomposition for Banach spaces has been defined. A characterization for approximative atomic decompositions has been obtained. Also, it has been proved that a Banach space E has an approximative atomic decomposition if and only if it possesses bounded approximation property. Further, sufficient conditions for the existence of approximative atomic decompositions in separable Banach spaces have been obtained. Finally, as an application of approximative atomic decompositions, it has been proved that if E and F are Banach spaces having bounded approximation property, then E x F also has bounded approximative property.