# Pythagoras's Theorem (Hilbert Space)

## Theorem

Let $H$ be a Hilbert space with inner product norm $\norm {\, \cdot \,}$.

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

Then:

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

## 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 $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \innerprod {f_i} {f_j}$ Linearity of Inner Product $\ds$ $=$ $\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin {cases} \innerprod {f_i} {f_i} & i = j \\ 0 & i \ne j \end {cases}$ 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.