Coefficients with respect to orthogonal-like decomposition systems

Properties of orthogonal-like decomposition systems are analyzed. It is shown in § 1 that such systems have most properties of orthogonal systems; the properties of the coefficients of the corresponding expansions are studied. A criterion enabling one to determine whether an orthogonal-like system is an orthogonal projection of an orthogonal system is established in § 2. In § 3 it is shown that Parseval's equality holds for a system satisfying certain natural conditions if and only if this is an orthogonal-like decomposition system.