Pythagoras's Theorem (Inner Product Space)

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $\norm \cdot$ be the inner product norm on $\struct {V, \innerprod \cdot \cdot}$.

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

Then:


 * $\ds \norm {\sum_{i \mathop = 1}^n f_i}^2 = \sum_{i \mathop = 1}^n \norm {f_i}^2$

Also see

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