A seminorm characterization of infinite Banach direct sums

In this paper, the notion of a \(\Delta \)-direct sum of a family of Banach spaces indexed by a set I, where \(\Delta \) is a union-closed subnet of \(\textsf{Fin}(I)\) (the family of all finite subsets of I), is introduced. A seminorm characterization of \(\Delta \)-direct sums and some results are presented. Necessary and sufficient conditions are found that a direct sum of a family of Banach spaces is a \(\Delta \)-direct sum. Elements of a direct sum of Banach spaces that are \(\Delta \)-sectionally convergent are introduced and studied. Examples of \(\Delta \)-direct sums and applications of \(\Delta \)-direct sums to Fourier analysis on compact groups are given.