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:


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

Note
If $H$ is $\R^2$ with the usual inner product, and $n=2$, this theorem reduces to the well-known Pythagoras's Theorem.