Pythagoras's Theorem (Inner Product Space)

Theorem
Let $H$ be a Hilbert space with inner product norm $\left\|{\cdot}\right\|$.

Let $f_1, \ldots, f_n \in H$ be pairwise orthogonal.

Then:


 * $\displaystyle \left\|{\sum_{i \mathop = 1}^n f_i}\right\|^2 = \sum_{i \mathop = 1}^n \left\|{f_i}\right\|^2$

Also see

 * Pythagoras's Theorem, the well-known instance of this theorem with $H = \R^2$ and the usual inner product.