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.
