In this paper, the notion of a Δ-direct sum of a family of Banach spaces indexed by a set I, where Δ is a union-closed subnet of Fin(I) (the family of all finite subsets of I), is introduced. A seminorm characterization of Δ-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 Δ-direct sum. Elements of a direct sum of Banach spaces that are Δ-sectionally convergent are introduced and studied. Examples of Δ-direct sums and applications of Δ-direct sums to Fourier analysis on compact groups are given.
