# Pythagorean Theorem (Hilbert Space)

## Contents

## 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:

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

## Proof

\(\displaystyle \left\Vert{\sum_{i \mathop = 1}^n f_i}\right\Vert^2\) | \(=\) | \(\displaystyle \left\langle{\sum_{i \mathop = 1}^n f_i, \sum_{j \mathop = 1}^n f_j}\right\rangle\) | $\quad$ Definition of Inner Product Norm | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \left\langle{f_i, f_j}\right\rangle\) | $\quad$ Linearity of Inner Product | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin{cases}\left\langle{f_i, f_i}\right\rangle & i = j\\ 0 & i \ne j\end{cases}\) | $\quad$ The $f_i$ are pairwise orthogonal | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \left\Vert{f_i}\right\Vert^2\) | $\quad$ Definition of Inner Product Norm | $\quad$ |

$\blacksquare$

## Also see

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

## Source of Name

This entry was named for Pythagoras of Samos.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $I.2.2$