Bessel’s Inequality

For simplicity, we adopt the following convention: X denotes a real unitary space, x, y, y1, y2 denote points of X, i, j denote natural numbers, D1 denotes a non empty set, and p1, p2 denote finite sequences of elements of D1. Next we state the proposition (1) Suppose p1 is one-to-one and p2 is one-to-one and rng p1 = rng p2. Then dom p1 = dom p2 and there exists a permutation P of dom p1 such that p2 = p1 · P and domP = dom p1 and rngP = dom p1. Let D1 be a non empty set and let f be a binary operation on D1. Let us assume that f is commutative and associative and has a unity. Let Y be a finite subset of D1. The functor f⊕Y yields an element of D1 and is defined as follows: (Def. 1) There exists a finite sequence p of elements of D1 such that p is one-toone and rng p = Y and f ⊕ Y = f ⊙ p. Let us consider X and let Y be a finite subset of the carrier of X. The functor SetopSum(Y,X) is defined as follows: