Let $H$ be an inner product space. If $x_1 ,x_2,\ldots , x_n \in H$ are pairwise orthogonal then
$$ ||\sum_{k=1}^{n} x_k||^2=\sum_{k=1}^{n} ||x_k||^2 $$If $\{x_1 ,x_2,\ldots , x_n\}$ is a orthonormal subset and $c_k\in \mathbb{R}$ 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.
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Author of the notes: Antonio J. Pan-Collantes
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