The Generalized Pythagorean Identity for Inner Product Spaces

Let $H$ be an inner product space. If $x_{1}, x_{2}, \dots, x_{n} \in H$ are pairwise orthogonal then

| | \sum_{k = 1}^{n} x_{k} | |^{2} = \sum_{k = 1}^{n} | | x_{k} | |^{2}

If ${x_{1}, x_{2}, \dots, x_{n}}$ is a orthonormal subset and $c_{k} \in R$ then it can be restated as

| | \sum_{k = 1}^{n} c_{k} x_{k} | |^{2} = \sum_{k = 1}^{n} | c_{k} |^{2}

If the orthonormal set is infinite we only have Bessel's inequality.