# 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:

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

## Proof

 $\ds \norm {\sum_{i \mathop = 1}^n f_i}^2$ $=$ $\ds \innerprod {\sum_{i \mathop = 1}^n f_i} {\sum_{j \mathop = 1}^n f_j}$ Definition of Inner Product Norm $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \innerprod {f_i} {f_j}$ since the Inner Product is linear in the first argument $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin {cases} \innerprod {f_i} {f_i} & i = j \\ 0 & i \ne j \end {cases}$ the $f_i$ are pairwise orthogonal $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \norm {f_i}^2$ Definition of Inner Product Norm

$\blacksquare$

## Source of Name

This entry was named for Pythagoras of Samos.