Journal of the
Korean Mathematical Society JKMS

In this paper, we form the basis of the abstract theory for constructing the K\"{u}nneth spectral sequence for a complex of Banach spaces. As the category of Banach spaces is not abelian, several difficulties occur and hinder us from applying the usual method of homological algebra directly. The most notable facts are the image of a morphism of Banach spaces is not necessarily a Banach space, and also the closed summand of a Banach space need not be a topological direct summand. So, we consider some conditions and categorical terms that fit the category of Banach spaces to modify the familiar method of homological algebra.

Keywords: K\"{u}nneth spectral sequence, Banach complex

MSC numbers: Primary 55T99; Secondary 18G40