# Pythagoras's Theorem (Hilbert Space)

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## Contents

## 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

\(\displaystyle \norm {\sum_{i \mathop = 1}^n f_i}^2\) | \(=\) | \(\displaystyle \innerprod {\sum_{i \mathop = 1}^n f_i} {\sum_{j \mathop = 1}^n f_j}\) | Definition of Inner Product Norm | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \innerprod {f_i} {f_j}\) | Linearity of Inner Product | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin {cases} \innerprod {f_i} {f_i} & i = j \\ 0 & i \ne j \end {cases}\) | The $f_i$ are pairwise orthogonal | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{i \mathop = 1}^n \norm {f_i}^2\) | Definition of Inner Product Norm |

$\blacksquare$

## 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) $I.2.2$